Apparatus for and method of pattern recognition and image analysis

ABSTRACT

A method of comparing an input pattern with a memory pattern includes the steps of loading a representation of said input pattern into cells in an input layer; loading a representation of said memory pattern into cells in a memory layer; loading an initial value into cells in an intermediate layers between said input layer and said memory layer; comparing values of cells in said intermediate layers with values stored in cells of adjacent layers; updating values stored in cells in said intermediate layers based on said step of comparing; and mapping cells in said memory layer to cells in said input layer.

RELATED APPLICATIONS

This application is a continuation-in-part (CIP) of U.S. patentapplication Ser. No. 10/144,754 filed May 15, 2002 which claims priorityto U.S. Provisional Patent Application Ser. No. 60/291,000, filed on May16, 2001.

FIELD OF THE INVENTION

The present invention is directed to methods of and systems forinformation processing, information mapping, pattern recognition andimage analysis in computer systems.

BACKGROUND

With the increasing proliferation of imaging capabilities, informationtransactions in computer systems increasingly require the identificationand comparison of digital images. In addition to, conventional viewabledigital images, other types of information, both viewable andnon-viewable, are subject to pattern analysis and matching. Imageidentification and pattern analysis/recognition is usually dependent onanalysis and classification of predetermined features of the image.Accurately identifying images using a computer system is complicated byrelatively minor data distorting the images or patterns resulting fromchanges caused when, for example, are shifted, rotated or otherwisedeformed.

Object invariance is a field of visual analysis which deals withrecognizing an object despite distortion such as that caused byshifting, rotation, other affine distortions, cropping, etc. Objectinvariance is used primarily in visual comparison tasks. Identificationof a single object or image within a group of objects or images alsocomplicates the image identification process. Selective attention, or“priming”, deals with how a visual object can be separated from itsbackground or other visual objects comprising distractions.

Current pattern recognition, image analysis and information mappingsystems typically employ a Bayesian Logic. Bayesian Logic predictsfuture events through the use of knowledge derived from prior events. Incomputer applications, Bayesian Logic relies on prior events toformulate or adjust a mathematical model used to calculate theprobability of a specific event in the future. Without prior events onwhich to base a mathematical model, Bayesian Logic is unable tocalculate the probability of a future event. Conversely, as the numberof prior events increases, the accuracy of the mathematical modelincreases as does the accuracy of the resulting prediction from theBayesian Logic approach.

Currently, two common paradigms accommodating some degree of distortion(i.e., image deformation) of a visual object under deformations;point-to-point mapping and high order statistics. Point-to-point, ormatching with shape contexts, achieves measurement stability byidentifying one or more sub-patterns with the overall patterns or imagesbeing compared. Once these sub-patterns are identified, the statisticalfeatures of sub-patterns are compared to determine agreement between thetwo images. Point-to-point mapping methodologies are further describedin “Matching with Shape Contexts” by Serge Belongie and Jitendra Malikin June, 2000 during the IEEE Workshop On Content-based Access of Imageand Video Libraries (CBAIVL). A second method of point-to-point mappingis what/where networks and assessments of lie groups of transformationsbased on back propagation networks. In this method an optimaltransformation is identified for a feature in a first image and is usedto compare the same feature in a second image. This approachdeconstructs the image into a sum or a multiplicity of functions. Thesefunctions are then mapped to an appropriately deconstructed imagefunction of a compared, or second image. What/where networks have beenused by Dr. Rajesh Rao and Dana Ballard from the Salk Institute in LaHoya, Calif. The point-to-point mapping techniques described attempt tomap a test or input image to a reference or target image that is eitherstored in memory directly or is encoded into memory. The point-to-pointapproach achieves limited image segmentation and mappings through theuse of a statistical approach.

In the high order statistical approach both the original input image andthe compare target image are mapped into a high dimensional space andstatistical measurements are performed on the images in the highdimensional space. These high order statistical measurements arecompared to quantity an amount of agreement between the two imagesindicative of image similarity. This approach is used by Support VectorMachines, High Order Clustering (Hava Siegelmann and Hod Lipson) andTangent Distance Neural Networks (TDNN). Support Vector Machines aredescribed by Nello Christianini and John Shawe-Taylor ISBN0-521-78019-5.

Both the point-to-point mapping and the high order statistics approachhave been used in an attempt to recognize images subject to varioustransformations due to shifting, rotation and other deformations of thesubject. These approaches are virtually ineffective for effectivelyisolating a comparison object (selective attention) from the backgroundor other visual objects.

In contrast to these two common paradigms, the human brain may comparetwo objects or two patterns using “insight” without the benefit of priorknowledge of the objects or the patterns. A Gestalt approach tocomparing objects or comparing patterns attempts to include the conceptof insight by focusing on the whole object rather than individualportions of the object. Gestalt techniques have not been applied tocomputer systems to perform pattern recognition, image analysis orinformation mapping. Gestalt mapping is further described in VisionScience-Photons to Phenomenology by Stephen E. Plamer, ISBN0-262-16183-4.

SUMMARY

According to one aspect of the present invention, a method of comparingan input pattern with a memory pattern comprises the steps of loading arepresentation of said input pattern into cells in an input layer;loading a representation of said memory pattern into cells in a memorylayer; loading an initial value into cells in an intermediate layersbetween said input layer and said memory layer; comparing values ofcells in said intermediate layers with values stored in cells ofadjacent layers; updating values stored in cells in said intermediatelayers based on said step of comparing; and mapping cells in said memorylayer to cells in said input layer.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows a flow diagram of one embodiment of the transformationengine data processing of the present invention;

FIG. 2 shows an embodiment of the present invention including a onedimensional numeric transformation engine which may be used to map timedependent curves;

FIG. 3A is a one dimensional, time dependent curve representing an inputpattern;

FIG. 3B is a one dimensional, time dependent curve representing a memorypattern;

FIG. 3C shows the one dimensional, time dependent curves representingsuperimposed input and memory patterns;

FIG. 3D shows the one dimensional, time dependent curves representingthe layers within a matrix during the early stages of top-down andbottom-up wave traversal;

FIG. 3E shows the one dimensional, time dependent curves representing anadvanced stage top-down and bottom-up wave traversal;

FIG. 4 shows the structure of a one-dimensional numerical transformationengine including a top-down and bottom-up wave propagation approach ofone embodiment of the present invention;

FIG. 5 shows a screen shot of a mapping between a stored signature andinput signature with corresponding points connected by line segments;

FIG. 6 represents another embodiment of the present invention which maybe used to implement selective attention filtering of noise;

FIG. 7 shows a two-dimensional hexagonal implementation of atransformation engine according to an embodiment of the invention;

FIG. 8 is an illustration of either the input short term or long termmemory in which metal electrodes that are located in the middle of eachpixel are each fed with a different alternating or constant voltage atdifferent frequencies;

FIG. 9 shows an illustration of a mathematical model of one embodimentof the transformation engine;

FIG. 10 shows how a disk filled of “−” signs may be mapped to a diskfilled with “+” signs in a comparison of images;

FIG. 11 shows the interaction between the input layer and the memorylayer within a transformation engine;

FIG. 12 shows one embodiment of a unit or cell used in an embodiment ofthe transformation engine;

FIG. 13 shows a spherical cell of a solid transformation engineincluding a colloid with suspended dipolar media units;

FIG. 14 shows a transformation engine including a liquid mediasuspending or including dipolar media units according to an embodimentof the present invention;

FIG. 15A shows the cyclic structure of the S, S+1, and S+2 nature of thethree layers of a dipolar transformation engine;

FIG. 15B shows the top layer, S, of the dipolar transformation engine;

FIG. 16 shows the orientation of the layers when the distance of eachvertex from the heights intersection is

$\frac{1}{\sqrt{3}}$

FIG. 17 shows a flow diagram of another embodiment of the transformationengine data processing of the present invention.

DETAILED DESCRIPTION

The following terms and their definitions are provided to assist in theunderstanding of the various embodiments of the present invention.

-   -   (1) Hierarchical Zooming as used in Multi-resolution Analysis        and further explained in “Introduction to Wavelets and Wavelet        Transforms”, A Primer, C. Sidney Burrus, Ramesh A. Gopinath,        Haitao Guo, Prentice Hall ISBN 0-13-489600-9 Chapter 2 page 10,        A Multi-resolution Formulation of Wavelet Systems and “a Wavelet        tour of signal processing”—Stephane Mallat ISBN 0-12-466606-X        page 221.    -   (2) Ebbing resonance, Relaxation process as used in Simulated        Annealing and further explained in Adaptive Cooperative Systems,        Martin Beckerman, ISBN 0-471-01287-4, Chapter 7, page 253, The        approach to equilibrium, 7.1.1 Relaxation and Equilibrium        Fluctuations.        -   See also “Simulated Annealing” at Chapter 4, page 108. In            connection with a transformation engine according to the            present invention, simulated annealing results in a gradual            reduction of the values of ‘R’ ‘K’ and ‘a’ over time.    -   (3) Intensity of Resonance means the value of the learning rate        or update rate 0<R<1 such that if R is close to 1 then the        “Resonance is Strong”, and for 128 layers when R>0.955 then the        Resonance is considered as “strong”.    -   (4) Multi-level matrix refers to a transformation engine in        which each cell/unit/pixel can hold more than one value at the        time and in which these different values are used at different        phases of the algorithm. For example, a one-dimensional        transformation engine stores (x,y) values in its        cells/units/pixels. An additional (x₁,y₁) vector can also be        stored.    -   (5) Euclidean distance (between (x,y,z) and (x₁,y₁,z₁)=square        root of ((x−x₁)²+(y−y₁)+(z−z₁)²).    -   (6) Far Differentials refer to Far Span k-Distance Connections        such that, given 4 values of 4 pixels, p1,p2,q1,q2 such that        p1,p2 are on one layer and q1,q2 are on an adjacent layer, let K        be the distance between the p1 and p2 and the same distance        between q1 and q2, then the Far Span k-Distance Connection        between the 4 values is (p2−p1)−(q2−q1), such that the value p1        is in pixel (i₁,j₁,S), the value p2 is in pixel (i₂,j₂,S), the        value q1 is in pixel (i₃,j₃,S+1) and the value q2 is stored in        pixel (i₄,j₄,S+1). K means that the square root of        (i₁−i₂)^(2+(j) ₁−j₂)₂ is K and that also the square root of        (i₃−i₄)²+(j₃−j₄)² is also K. In a one dimensional transformation        engine instead of having two indices (i, j) in each layer we        have only one index.    -   (7) Energy, including the phrase “a more complex energy        function”, refer to a function minimized by a transformation        engine according to an embodiment of the invention. The Energy        function may be expressed by following integral in a two        dimensional transformation engine,

∫_(x,)∫_(y,)∫_(z)(∇P ⋅ ∇P + K(∇∇P ⋅ ∇P) ⋅ (∇∇P ⋅ ∇P) + Optional) 𝕕x⋅ 𝕕y⋅ 𝕕z

-   -    given pixel (x,y) at layer z for a scalar energy P.        -   If each pixel holds a vector V instead of a scalar P then            the gradient of P is replaced by the vector V and the second            gradient of P is replaced by the gradient of V.        -   Then the energy/Action integral takes the form,

∫_(x,)∫_(y,)∫_(z)(V ⋅ V + K(∇V ⋅ V) ⋅ (∇V ⋅ V) + Optional) 𝕕x⋅ 𝕕y⋅ 𝕕z

-   -   -   The General Relativistic tensor form of the energy function            may be expressed as:

$\int_{\Omega}( {P,_{m}P,_{k}{g^{mk}{\quad{+ {\quad{\quad{{Kg}^{Lk}( {P,_{j}{{g^{jn}( {P,_{k},_{n}{{- \begin{Bmatrix}u \\{kn}\end{Bmatrix}}P},_{u}} )}( {P,_{j}{{{g^{jn}( {P,_{L},_{n}{{- \begin{Bmatrix}u \\{Ln}\end{Bmatrix}}P},_{u}} )}\sqrt{- g}\ {\mathbb{d}\Omega}} = {\int_{\Omega}( {P,_{m}P,^{m}{+ {KP}},_{n}P,_{k}{;^{n}P},_{m}P,^{k}{;{m\sqrt{- g}{\mathbb{d}\Omega}}}} }}} }} }}}}}}} $

-   -   -    where g_(km) is Einstein's metric tensor and the term

$\{  \quad\begin{matrix}u \\{kn}\end{matrix} \} $is the Christoffel symbol. The General Relativistic energy function is amodel of the cosmos as a transformation engine in equilibrium.

-   -   (8) Distance function is a function D that receives two points        and calculates a value. For example the two points (x,y),        (x₁,y₁) and the functions D((x, y),(x₁,y₁))=e^((x−x) ¹ ⁾ ²        ^(+(y−y) ¹ ⁾ ² or D((x,y),(x₁,y₁))=√{square root over        ((x−x₁)²+(y−y₁)²)}{square root over ((x−x₁)²+(y−y₁)²)}    -   (9) Radial basis decay function is a function g that is reduced        with the distance from a given point (x₀,y₀). For example        g(x,y)=e^(−(x−x) ⁰ ⁾ ² ^(−(y−y) ⁰ ⁾ ²    -   (10) Convolution Scale is a convolution with a Radial basis        decay function such that the distance from the point x₀,y₀        (refer to definition 9 above) is divided by a constant s        terminal the scale factor:

${g( {x,y} )} = {\mathbb{e}}^{\frac{{- {({x - x_{0}})}^{2}} - {({y - y_{0}})}^{2}}{s^{2}}}$

-   -   -   This function is the same as in 9 but the distance √{square            root over ((x−x₁)²+(y−y₁)²)}{square root over            ((x−x₁)²+(y−y₁)²)} is divided by S. The square distance is            divided by S².

    -   (11) Pattern degeneracy, in connection with symmetry, if the        derivative of a potential function p vanishes in a domain D then        P is degenerated in D; and in connection with a multidimensional        manifold, such as the 4 dimensional manifold        (x,y,z,t,p(x,y,z,t)), wherein p depends on the other        coordinates, one can consider the Gaussian curvature of the        manifold, such that, if the curvature is 0 in the domain D, then        the potential p is said to be degenerated on D. The last        definition is appropriate only if x,y,z,t define and Euclidean        space.

    -   (12) Potential is a function that depends on coordinates and is        a scalar value.        -   Scalar value means that the function calculates a            directionless numeric value and not a vector. Thus, a            generic potential is a function that calculates a value with            no dimensions, such as the coulomb potential, 2 inches is a            value with the dimension of length,

${p( {x,y,z,t} )} = \frac{q}{\sqrt{( {x - x_{0}} )^{2} + ( {y - y_{0}} )^{2} + ( {z - z_{0}} )^{2}}}$

-   -   (13) Hebbian learning and Hebbian learning rate means that, if a        Unit(s,t) is updated towards a new data vector V then the new        value of Unit(s,t) will be, (1−R) Units(s,t)+R·V such that R<1        where R is the Hebbian learning rate.    -   (14) Updates refer to an update of a value or vector v towards        the value or vector u, the value r*(u−v) such that r is the        resonance rate added to v, or v(n+1)=v(n)+r*(u(n)−v(n)).    -   (15) Resonance rate refers to the Hebbian rate at which a unit        is updated towards a “winning” unit in an adjacent layer. The        resonance rate ‘r’ or ‘R’ is a number between 0 and 1 not        including 0 and 1, i.e., 0<r<1.    -   (16) Move Towards Winner refers to a competition, such that,        suppose that n vectors V₁,V₂, . . . V_(n) are checked against a        vector U to determine which is closest to U in Euclidean        coordinates. Further, suppose that |V_(j)−U| is the minimum norm        for some 0<j<n=1. Then if the value r* (V_(j)−U) such that 0<r<1        is added to U then U is said to Move Towards the Winner V_(j).    -   (17) Winner Takes All refers to a competition, such that,        suppose that n vectors V₁,V₂, . . . V_(n) are checked against a        vector U which is closest to U in Euclidean coordinates.        Further, suppose that |V_(j)−U| is the minimum norm for some        0<j<n+1, then if the value r*(U−V_(j)) such that 0<r<1 is added        to V_(j) then Vj is said to be the Winner That Takes All.    -   (18) Ebbing Resonance refers to a relaxation process, wherein        “relaxation” is used with the same meaning as in non-linear        optimization theory. Relaxation typically involves a gradual        reduction parameters such as heat, noise, peak to peak        alternating current amplitude, speed, learning rate, electric        field intensity, etc. The word “resonance” is used in the same        way that it is used in ART (Adaptive Resonance Theory) as        articulated by Professor Stephen Grossberg.    -   (19) Euclidean Distance is the distance between a vector V and a        vector U that is the norm of the difference V−U in R^(n),

${{V - U}} = \sqrt{\sum\limits_{i = 1}^{n}\;( {V_{i} - U_{i}} )^{2}}$such that i is the index of the i^(th) dimension of the vectors V and U.

-   -   (20) Far Differential refers to a Far Span K Distance        Connection, such that the Far Differential−Far Span K Distance        Connection between a Units(s,t) and Unit(s,t+k) is        Unit(s,t+k)−Unit(s,t) and between Unit(s,t−k) and Unit(s,t) it        is Unit(s,t−k)−Unit(s,t).    -   (21) Hexagonal Coordinates refers to a hexagonal matrix having        the following indices, for an even row, 0, 2, 4, 8, . . . , 2n.        For an odd row, 1, 3, 5, 7, . . . 2n+1. For example the        following unit is valid Unit(s,30,30) and Unit(s,30,31) is        invalid. Indices are used like the row and column indices in        matrices in Linear Algebra.    -   (22) Hexagonal Neighbors (two-dimensional) defined such that the        Hexagonal neighbors of the unit Unit(s,w,t) are Unit(s,w+1,t+1),        Unit(s,w+1,t−1), Unit(s,w−1,t+1), Unit(s,w−1,t−1),        Unit(s,w,t+2), Unit(s,w,t−2).    -   (23) Hexagonal Far Differentials refers to Far Span K Distance        Connections (two-dimensional), such that for a valid unit        Unit(s,w,t) there are 6 such Far Differentials as follows,        Unit(s,w,t)−Unit(s,w+k,t+k), Unit(s,w,t)−Unit(s,w+k,t−k),        Unit(s,w,t)−Unit(s,w−k,t+k), Unit(s,w,t)−Unit(s,w−k,t−k),        Unit(s,w,t)−Unit(s,w,t+2k), Unit(s,w,t)−Unit(s,w,t−2k).    -   (24) ‘k’ Parameter is defined by Far Differential−Far Span K        Distance Connection and Hexagonal Far Differentials−Far Span K        Distance Connections.    -   (25) ‘a’ Parameter refers to a number that changes during the        ebbing resonance numeric process. Instead of calculating the        distance between the units U(s,t) and U(s,t−1) as |U(s,t)−U(s,        t+1)| the distance function uses Far Differentials as follows,        -   Distance(U(s,t), U(s+1,t+1))=|U(s,            t)−U(s+1,t+1)|+a*|U(s,t+k)−U(s,t)−U(s+1,t+1+k)+U(s+1,t+1)|+a*|U(s,t−k)−U(s,t)−U(s+1,t+1−k)+U(s+1,t+1)|.        -   Distance(U(s,t), U(s+1,t−1))=|U(s,            t)−U(s+1,t−1)|+a*|U(s,t+k)−U(s,t)−U(s+1,t−1+k)+U(s+1,t−1)|+a*|U(s,t−k)−U(s,t)−U(s+1,t−1−k)+U(s+1,t−1)|.        -   Distance(U(s,t),            U(s+1,t))=|U(s,t)−U(s+1,t)|+a*|U(s,t+k)−U(s,t)−U(s+1,t+k)+U(s+1,t)|+a*|U(s,t−k)−U(s,t)−U(s+1,t−k)+U(s+1,t)|.        -   Using the Hexagonal Far Differential obtains a sum of 7            Euclidean terms in which six of them are multiplied by the            ‘a’ parameter.    -   (26) Convolution (of a scalar or a vector function G which is        defined on the domain D with a function f at the point X) is        defined as,

Convolution(G, X) = ∫_(D)G(Z) * f(Z − X) 𝕕Z

-   -   (27) Hierarchical Zooming means an Ebbing Resonance process in        which the ‘k’ parameter either in the one-dimensional numeric        Transformatron or in the Two Dimensional Hexagonal        Transformatron Far Differentials is gradually reduced. The term        Hierarchical Zooming also refers to a convolution with Radial        Basis Decay function such that the convolution integral takes        the form,

${{Convolution}( {G,X,t} )} = {\int_{D}{{G(Z)}*{f( \frac{Z - X}{R(t)} )}\ {\mathbb{d}Z}}}$

-   -   -   In this formula, R(t) is called Radial Scale and it is            gradually reduced with the time t.

    -   (28) Convolution Scaling refers to a function R(t) which is        defined in the Hierarchical Zooming.

    -   (29) Radial Basis refers to a function f(Z) is considered as a        Radial Basis function if and only if f(Z)=u(|Z|) such that u is        a function that depends only on the norm of Z.

    -   (30) Radial Basis Decay refers to a function of the form,        f(Z)=u(|Z|) and that the following conditions hold,

∫_(D)f(Z) 𝕕Z < ∞⋀(Z → ∞) ⇒ f(Z) → 0.

-   -   -   If the domain D is finite then the first condition holds.

    -   (31) Clue Oriented Mapping [of two patterns f, g] is illustrated        as follows. Let f and g denote bounded analytic real functions        that are defined on open connected domains D_(f) and D_(g) such        that there is a map F: D_(f)→D_(g), ∀pεD_(f)(p),        Determinant(F)≠0 and F is globally onto and one to one. If the        domains are finite sets in R^(n) then we are done otherwise        there is another requirement that the source of each compact set        in D_(g) should also be compact. This mapping F is called Proper        in the mathematics (theory of Differential Topology).        -   We also require that ∀pεD_(f)(p),g(F(p))=f(p). A proper map            that minimizes the following ∫C((K_(F(Z)) ^(t)−K_(z)            ^(t))²)_(dz) such that K is the Gaussian Curvature of the            manifolds D_(f),f and D_(g),g respectvely is defined Clue            Oriented. C is a function that defines how the system            responds to curvature mismatches. Example for C is            C(x)=log(1+k).

    -   (32) Degenerated Pattern refers to pattern degeneracy, wherein a        degeneration pattern is a function f that is defined on a domain        D_(f) such that there exists an open set Q such that the        Gaussian curvature K of the manifold D_(f),f on Q is 0.

    -   (33) Degree of Freedom refers to a condition in which more than        one possible solution can be reached.

    -   (34) Phase Lag refers to the time from the moment a particle is        placed in an electric field until it gains most of its induced        dipole, say 0.9 of the final dipole.

One embodiment of present invention provides a “transformation engine”that incorporates a Gestalt approach to image comparisons, patternrecognition and image analysis. The present invention may be used tocreate a point-to-point mapping or a point-to-point map between an inputpattern and a stored pattern (for convenience of explanation togetherreferred to as two visual objects although applicable to other datahaving or representative of a pattern). This point-to-point mapping maybe implemented by performing a transformation between the two visualobjects and creating a multi-layered gradual transition between theinput pattern and the stored or memory pattern. The point-to-pointmapping is performed by Resonance as defined hereinabove. Thus,Resonance is used to construct a transformation between the two visualobjects using a multi-layered gradual transition. This multi-layeredentity according to one embodiment of the invention constitutes a matrixformed of a plurality of layers. Each layer, within the multi-layeredapproach, maintains the topological features of each of the two visualobjects. That is, intermediate layers maintain and exhibitcharacteristics common to the two visual objects such that significantfeatures of the pattern are recognizable. As the number of layersincreases, the difference between two adjacent layers is reduced. Whilethis embodiment of the present invention does not explicitly minimizethe global “energy” function, the embodiment may achieve a Gestaltistenergy minimization of a transformation between the two visual objects.

Energy minimization is a consequence of the structure of thetransformation engine of the embodiment. Energy minimization may beeffected by an information “smearing” procedure, which causes amigration of data through the matrix to achieve a minimized energypotential between layers. Transitions between layers may be created byproviding “waves” of modifying functions. Two waves, a top down wave anda bottom up wave, may be used to create a Gestalt transformation betweenthe two visual objects. The input pattern may be stored in the inputlayer or the short term memory (STM) layer and the stored “reference” or“target” pattern may be stored in the memory layer or the long termmemory (LTM) layer. The transformation engine of the present inventionachieves the point-to-point mapping between the two visual objectswithout either directly or indirectly calculating statistical values.Once the point-to-point association between the two visual objects isachieved, stable statistical measurements may be calculated to determinethe amount of agreement between the two visual objects. This embodimentof the present invention may be used for comparing patterns of visualobjects (a comparison mode) and for isolating an object from thesurrounding background or other objects (selective attention task).

The transformation engine of the present invention includes or operatesin two phases. In the first phase the transformation engine builds a mapbetween the LTM (referenced or target image) pattern and the STM (inputpattern under examination) pattern. In the second phase, thetransformation engine uses “vibration” waves to turn the map createdinto either an explicit point-to-point mapping or into an explicitpattern recognition engine. The vibration waves cause the intermediatelayers to maintain a low energy state while encouraging conformance toLTM and STM patterns based on distance from such.

The map created by the transformation engine may be created in a numberof ways, each of which is included in the present invention. In oneembodiment the map may be created by digital means, e.g., a digitalcomputer. In another embodiment the map may be created by physicalmeans, i.e., incorporating a media responsive to patterned conditions soas to provide a transition between patterns, such as by placingelongated pieces of conducting material within an electric field suchthat the conducting material becomes aligned within the electric field.Other means of creating the mapping from the LTM pattern to the STMpattern are also included within the present invention.

During the second phase of operation of the transformation engine,vibration waves cause the intermediate values in the intermediatelayers—between the LTM and the STM layers to oscillate which preventsthe transformation engine from permanently storing local minimum of thecorresponding Action Integral within the intermediate layers. In otherwords, the transformation engine is aimed at eliminating the Variationof the (static) Action Integral:

0 = δ∫_(x,)∫_(y,)∫_(z)(∇P ⋅ ∇P + K(∇∇P ⋅ ∇P) ⋅ (∇∇P ⋅ ∇P) + Optional) 𝕕x⋅ 𝕕y⋅ 𝕕zWhere LTM=P(X,Y,0) and STM=P(X,Y,1).

The elimination of the variation of the Action Integral means that thepotential p maintains up to second order derivatives as the pattern thatis encoded by p changes from one layer to the next adjacent layer.Convergence in a digital implementation where neither a real potentialnor its gradient (the induced vector field) may be achieved throughdigital means, i.e., by use of a digital representation of the vectorfield using for example, a mathematical model to be explained below. Ina preferred embodiment, the transformation engine forces values tochange in the intermediate layers through the use of a learning rate (R)that is typical to concurrent neural networks approach. In thisembodiment, if P(S) represents the value stored in an intermediatelayer, and it is desired to change P(S) into a new value W then thelearning rate 0<R<1 defines the learning paradigm,P_(new)(S)=(1−R)*P_(old)(S)+R*W. If R is close to 1 then the learningprocess is rapid but relatively unstable. The reduction of thisparameter from 0.995 to 0.835 with the number of layers is 128 alongwith the reduction of the Far Span k-Distance Connections (a) is a wayto minimize the Action Integral. In a “real” physical system, i.e., onerelying on some physical property of a component of the system, neitherthe learning rate nor the Far Span K-Distance Connection would benecessary.

A digital implementation of the comparison mode of the present inventionwill be described first.

FIG. 1 shows a flow diagram of one embodiment of a transformation enginedata processing of the present invention. In Step 101 an input patternis loaded into the input layer or “short term memory” (STM) layer. Notethat the input or STM layer may be represented by or stored in eitherthe top most layer of the matrix or the bottom most layer of the matrix.For ease of explanation, the input layer will be defined as the top mostlayer of the matrix. In Step 102 the memory pattern is loaded into thememory layer or long term memory (LTM) layer. The memory or LTM layermay also be the bottom-most layer or the top-most layer opposite the STMlayer. That is, the memory layer should be at the opposite end of thematrix from the input layer. For ease of explanation, the memory layerwill be defined as the bottom most layer of the matrix.

In Step 103 data values representing some pattern to be identified(e.g., an input image) are stored in the input layer and copied intoeach level of the top or upper half of the multiple layers. Similarly,in Step 104 data values representing some previously stored pattern arestored in the memory layer and copied into each level of the lower orbottom half of the matrix. Steps 103 and 104 provide the cells of theinner layers of the matrix with initial data values to initialize thematrix. In Step 105 a top-down “wave” and a bottom-up :wave: arepromulgated through the multi-layer matrix as further described below.When Step 105 is performed during a comparison mode, the values in theinput layer and the memory layer are held constant, but the initialvalues stored in each of the inner layers of the matrix may be adjustedto provide a gradual transition of pattern data between layers. Duringthis step the “intensity” of the waves is gradually reduced, i.e., themagnitude of change between layers is gradually reduced, as is the spanof data checked as the method progresses towards convergence and energyminimization. This reduction in the intensity of the waves occurs in thefirst phase of the transformation engine when a map is constructedbetween the LTM and the STM pattern. Ebbing Resonance is also performedin Step 105 as further described below. The values in each of the layersare held constant in Step 106 and a one-to-one mapping is performed inStep 107 between the input layer and the memory layer.

Various methodologies may be used to perform this one-to-one mappingincluding sequentially sending a single wave for each input cell anddetermining, based on the single wave, the best matching memory cell. Inthis case, for each pixel of the STM input board/layer represented by(i, j) the value of the pixel (i,j) is oscillated or varied over apredefined range while the value for all other pixels (i.e., dataelements of the pattern) of the STM board are held constant. Thisprocess is repeated serially for each one of the pixels of the STMboard. The best matching memory cell using this methodology is thememory cell which receives the strongest vibration wave from the singlewave applied to the input cell. Once the best matching memory cell isidentified, the corresponding input cell is mapped to the memory cell.Alternatively, a coherent vibration wave may be sent from all of thecells in the input layer and spectral analysis of the wave may beperformed to determine the associations between input cells and memorycells. This second methodology creates a vibration wave pattern in eachof the memory cells which is a superposition of small ripple vibrationwaves. By measuring and performing spectral deconstruction on theresulting pattern in the memory layer, a Bayesian pattern may beidentified.

FIG. 2 shows an embodiment of the present invention which includes a onedimensional numeric transformation engine which may be used to map timedependent curves. FIG. 3A is a one dimensional, time dependent curvewhich is represented by input pattern 301. FIG. 3B is a one dimensional,time dependent curve which is represented by memory pattern 302. Asdiscussed in connection with Step 101, initially, data representing theinput signal curve is loaded into the input layer 201, which isrepresented by the “y” coordinates (e.g., sampling points or times) fromFIG. 3A which are thereby contained within layer 201. Preprocessing mayalso be performed on the values prior to storage in input layer 201. Thememory curve is loaded into memory layer 202 as discussed in connectionwith Step 102 and as shown in layer 202 of FIG. 2. The “y” coordinatesfrom FIG. 3B are contained in layer 202. Preprocessing may also beperformed on the values before they are stored within memory layer 202.Initially, each of the cells of layers 203 and 204 contain the samevalues in each of the respective cells of the layers as layer 201 asshown in FIG. 2 and as described in Step 103. Similarly, as shown inlayers 205 and 206, the “y” coordinates from the cells in memory layer202 are copied into the respective cells for layers 205 and 206 and asdescribed in Step 104. FIG. 3C shows the one dimensional, time dependentcurves which represent input pattern 301 and memory pattern 302superimposed. As can be seen, while the general form of the two patternsor signals are similar, each differs from the other in specifics.

Once each layer of the matrix stores initial values, Step 105 isperformed and both top-down and bottom-up waves are passed alternatelythrough the matrix. The top-down and bottom-up waves may modify thevalues stored in each of the cells to form a mapping between layers.(Note that, in a comparison mode, the values in the cells in the inputlayer and the memory layer remain constant through the comparison.) Alsoin Step 105, an ebbing resonance process is applied to the multi-levelmatrix. The ebbing resonance process is a relaxation process thatachieves convergence via hierarchical zooming. In a preferredembodiment, both the intensity of the resonance and the radius at whichdifferences between a cell and its neighboring cells are checked andgradually reduced. The complete ebbing resonance uses convolutions andtime varying patterns in a top-down and a bottom up approachalternately. The convolutions, as described more fully below, areinstrumental in overcoming energy pits and process degeneration around,for example, local minimum.

FIGS. 3D and 3E illustrate data stored in intermediate layers as themethod progresses toward a possible mapping, the intermediate layersproviding fiber resolution as each series of waves are propagated andconvergence is achieved.

FIG. 4 shows the structure of a one-dimensional numerical transformationengine of one embodiment of the present invention. transformation engine401 of FIG. 4 may hold the value of (x,y) representing discrete samplepoints of a time dependent curve by storing “x” amplitude value in arrayat index value “y” representative of sampling time. As previouslydescribed, in Step 105 two waves are present a top-down wave and abottom-up wave. In a preferred embodiment, the top-down wave and thebottom-up wave operate alternatively on the multi-level matrix totransform the data stored in the intermediate matrices toward a minimumenergy convergence. For ease of explanation, the operation of thetop-down wave and the bottom-up wave will be explained independently,and then an explanation will be given as to how these two waves worktogether to adjust the values of the cells contained within the innerlayers of the multi-layer matrix. Also for ease of explanation,comparisons between the layers in FIG. 4 will use terms such as layerabove and layer below, referring to the positional orientation of thelayers in FIG. 4 as illustrated as horizontal rows of the matrix. Forthis explanation an adjacent layer below is defined to be, with respectto FIG. 4, a layer which is positioned immediately below the currentlayer with no intermediate layer. Similarly, for this explanation, anadjacent layer above is defined to be, again with respect to FIG. 4, alayer which is positioned immediately above the current layer. Valueswithin cells may also change during Step 105 from a present value to afuture value.

At the completion of Step 104 the present value of each cell is equal tothe initial cell value as described previously, i.e., to the valuestored in the closest top and bottom layer. Additionally, thedetermination of the future value of a cell is dependent on the positionof the cell within the layer. For interior cells, those cells which arenot positioned at the start or the end of the layer, the future value ofthe cell is dependent on a number of cells, e.g., three other cellscontained in nearly positions of the adjacent row according to a onedimensional transformation engine as described below. For end cells,those cells in the first or last position of a layer, the future valueof the cell is dependent on one other cell in a one dimensionaltransformation engine as described below. Note that the dependency of acell on three other cells represents one embodiment of the presentinvention and that other dependencies are within the scope of thepresent invention.

For the bottom-up wave, the future value of an interior cell isdetermined by checking the present value in three closest cells in theadjacent layer above. In one embodiment of the present invention, whenthe multi-level matrix is operating in a comparative mode, the values ofthe top most row (the input layer) are not adjusted and continue tocontain values representing the input pattern. Processing of thebottom-up wave therefore begins in the second layer. For each interior(i.e., non-end) cell in the second layer, the three closest cells in thefirst layer are examined to determine the cell in the first layer whichhas the value closest to the current value of the cell in the secondlayer. If “t” represents the cell index/location within a layer, and “s”represents the layer, Unit (s, t) may be used to represent the cell (orunit) in layer “s” in position “t”. Using this convention, the referencefor cell 402 in FIG. 4 would be Unit (2, 3). This convention is based onthe layers being numbered from 1 to 6 from top to bottom and the cellsbeing numbered from 1 to 8 from left to right of FIG. 4. In general, forthe bottom-up wave, each unit (s, t) examines unit (s−1, t−1), unit(s−1, t) and unit (s−1, t+1). For each of these units, a unit containinga value of x and y which is closest to the present values of x and y inEuclidean distance is identified. Note that the Euclidean distancebetween a vector V and a vector U is calculated as follows:

${{{between}\mspace{14mu}{two}\mspace{14mu}{points}\mspace{14mu} p\mspace{14mu}{and}\mspace{14mu} q} = \sqrt{\sum\limits_{i = 1}^{N}\;( {p_{i} - q_{i}} )^{2}}}\mspace{14mu}$

In the bottom-up wave, once the value for each cell in a layer has beendetermined using the three cells in the adjacent layer above, the valuesfor the cells of the adjacent layer below are calculated. In otherwords, for the bottom-up wave, once the values for the cells in layer 2are determined using the values of the cells in layer 1, the value forthe cells in layer 3 are determined using the values of the cells inlayer 2. This process continues until the values stored in the cells oflayer 4, are used to update the values of the cells in the next to thebottom layer (layer 5 in FIG. 4).

The top-down wave operates similarly. For the top-down wave, the futurevalue of the cell is determined by examining the three closest cells inthe adjacent layer below. In one embodiment of the present inventionwhen in comparison mode, the values of the lowest-most row (the memorylayer) are not adjusted and continue to contain values representing thememory pattern. The top-down wave therefore begins in the second fromthe bottom layer (layer 5 of FIG. 4). For each cell in layer 5, thethree closest cells in layer 6 (the layer immediately below layer 5) areanalyzed to determine the value in the cell which has the value closestto the current value of the cell in the layer 5. In general, for thetop-down wave, each unit (s, t) checks unit (s+1, t−1), unit (s+1, t)and unit (s+1, t+1). For each of these units, a determination is made asto which of the units contains a value of x and y which is closest tothe present values of x and y in the same manner as previouslydescribed. In the top-down wave, once the value for each cell in a layerhas been determined using the adjacent layer below, the values for thecells of the adjacent layer above are calculated. In other words, forthe top-down wave, once the values for the cells in layer 5 aredetermined using the values of the cells in the memory layer (layer 6),the value for the cells in layer 4 are determined using the values ofthe cells in layer 5. This process continues until the values stored inthe cells of the layer 3 are used to update the values of the cells inlayer 2.

As described, the bottom-up wave travels from the layer below the inputlayer (or the second layer in FIG. 4) to the layer above the memorylayer (or the fifth layer of FIG. 4). Similarly, the top-down wavetravels from the layer above the memory layer (the fifth layer of FIG.4) to the layer below the input layer (the second layer of FIG. 4). Asdescribed, both the bottom-up and the top-down waves will pass throughthe middle layers of the multi-layer structure. In a preferredembodiment, the bottom-up and top-down waves are applied to middlelayers alternatively. For example:

-   -   A) A bottom-up wave is applied to layer 2. During this        application of the bottom-up wave, values in layer 2 will be        updated using values in layer 1.    -   B) After the bottom-up wave is applied to each cell in layer 2,        a top-down wave will be applied to layer 5. During this        application of the top-down wave, values in layer 5 will be        updated using values in layer 6.    -   C) After the top-down wave is applied to each cell in layer 5,        the bottom-up wave is applied to layer 3. During this        application of the bottom-up wave, values in layer 3 will be        updated using values in layer 2.    -   D) After the bottom-up wave is applied to each cell in layer 3,        the top-down wave will be applied to layer 4. During this        application of the top-down wave, values in layer 4 will be        updated using values in layer 5.    -   E) After the top-down wave is applied to each cell in layer 4,        the bottom-up wave will be applied to layer 4. During this        application of the bottom-up wave, values in layer 4 will be        update using values in layer 3.    -   F) After the bottom-up wave is applied to each cell in layer 4,        the top-down wave will be applied to layer 3. During this        application of the top-down wave, values in layer 3 will be        update using values in layer 4.    -   G) After the top-down wave is applied to each cell in layer 3,        the bottom-up wave will be applied to layer 5. During this        application of the bottom-up wave, values in layer 5 will be        updated using values in layer 4.    -   H) After the bottom-up wave is applied to each cell in layer 5,        the top-down wave will be applied to layer 2. During this        application of the top-down wave, values in layer 2 will be        update using values in layer 3.

This alternating process completes one application of the bottom-up andtop-down waves in the 6 row matrix of the present example. Note thatbottom-up and the top-down waves must reach the middle layers inintermittent order. If there are n=2m layers then in one cycle thebottom-up wave reaches layer m first and only then does the top-downwave reach the m−1 layer. In the following cycle the top-down wavereaches the m−1 layer first and only then does the bottom-up wave reachthe m layer.

As described, each interior cell or unit is compared to three othercells or units in either a layer above (for the bottom-up wave) or alayer below (for the top-down wave). Thus, the cell is compared to thet−1, t and t+1 cells in the corresponding layer. For end cells eitherthe t−1 or the t+1 cell is unavailable. For the t=1 cells, the t−1 cellin the layer above or below is unavailable and for the t=maximum (t=8 inFIG. 4) the t+1 cell in the layer above or below is unavailable. Inthese circumstances, a different method of updating the value of the endcell is required. In these instances the Far Differentials−Far Span KDistance−span connections cannot be used either. For the end cells, Unit(s, 1) or Unit (s, m) in the bottom-up wave the cell value is set equalto the value of Unit (s+1, 1) or (s+1, m) respectively. For the endcells, Unit (s, 1) or Unit (s, m) in the top-down wave the cell value isset equal to the value of Unit (s−1, 1) or (s−1, m) respectively.

Note that using simple Euclidean distance for updating Unit (s, t)cannot guarantee convergence to a global minima of the sum of squaredistances. A more efficient method of achieving the global minimum is byuse of a Far Differential−Far Span K Distance Connection. The FarDifferential−Far Span K Distance Connection describes the metricsnecessary to achieve a global minimum. This procedure minimizes a morecomplex energy or distance function in order to implement complex visualcomparison tasks. The Far Differential−Far Span K Distance Connectionbetween Unit (s, t) and Unit (s, t+k) is Unit (s, t+k)−Unit (s, t).Similarly, the Far Differential−Far Span K Distance Connection betweenUnit (s, t−k) and Unit (s, t) is Unit (s, t−k)−Unit (s, t).

FIG. 3D shows the one dimensional, time dependent curves which representthe layers within the matrix during the early stages of the applicationof the top-down and bottom-up waves. Similarly, FIG. 3E shows the onedimensional, time dependent curves which represent an advanced stage ofthe application of the top-down and bottom-up waves.

There is a gradual reduction in the intensity of the changes in thevalues stored in the cells during the application of the top-down andbottom-up waves and an ebbing resonance stage is reached. EbbingResonance is a relaxation process. The word “relaxation” as used hereinhas the same meaning as in non-linear optimization theory. Here,relaxation is accomplished by gradually reducing the Far Span K DistanceConnections. While this reduction alone is insufficient to guaranteeconvergence, it provides additional stability to the procedure. In orderto guarantee convergence using a full numeric solution, a power oflogarithmic Radial Basic Decay function is used in a HierarchicalZooming convolution and is applied to the input/STM layer and thememory/LTM layer simultaneously. In addition, the Convolution scale isalso gradually reduced.

The Far Span K Distance Connections modifies the Euclidean distance thatis used by the top-down and bottom-up waves. For example, applicationfor Far Span K Distance Connections causes the Euclidean distance|U(s,i)−U(s,i+1)| to becomeDistance(U(s,i+1))=|U(s,i)−U(s+1,i+1)|+a*|U(s,i+k)−U(s,i)−U(s+1,i+l+k)+U(s+1, i+1)+a*|U(s, i−k)−U(s,i)−U(s+1, i+1−k)+U(s+1, i+1)|.During the ebbing resonance, the real number “a” is also graduallyreduced with “k” and with the radius of the Hierarchical Zoomingconvolution. If a convolution tool is not available and the values inthe cells or units are kept constant during the Ebbing Resonance,convergence may still be obtained by the use of a very high Resonancerate (0.995 for 128 layers) and gradually reducing the resonance rate,the “k” parameter and the “a” parameter. This procedure allows theresonance rate to be gradually reduced during Hierarchical Zooming.

Once the Ebbing Resonance process has been completed, update waves maybe used to carry information between the layers of the matrix. Forexample, an update wave may be used to propagate index relatedinformation from the input layer, through each of the intermediatelayers to the memory layer to create a point-to-point mapping betweencells in the input layer and cells in the memory layer. Top-down andbottom-up update waves may be applied to the matrix to determinepoint-to-point mappings. In a one-dimensional transformation engine,each unit or cell checks three units in an adjacent layer (the layerabove for bottom-up and the layer below for the top-down). One of thethree units in the adjacent layer holds a value of x, y which is closestto the values of x, y of the current unit in Euclidean distance. Anintrinsic index, which represents the unit in the adjacent cell whichhad the closest value, is then stored in the current unit. This processis repeated for each cell in the layer, and for each layer in thematrix. After each cell of each layer has been examined and an intrinsicindex has been stored, the intrinsic index propagates from the inputlayer to the memory layer and vice versa.

The final mapping is influenced by the number of layers contained in thematrix and the pattern degeneracy. Here a degenerated pattern is afunction f that is defined on a domain D_(f) such that there exists anopen set Q such that the Gaussian curvature K of the manifold D_(f)f onQ is 0.

FIG. 5 shows a mapping between a stored signature 501 and inputsignature 502 with corresponding points connected by line segments. Thebottom signatures include intermediate layer information. Note thatreflex points are maintained without a requirement for their independentidentification.

FIG. 6 represents another embodiment of the present invention which maybe used to perform selective attention so as to identify a predeterminedobject in amongst other objects and/or a noisy environment. In an imagecontext, as previously described, selective attention includes theability to isolate a visual object from a background or other objects.In the comparative mode of the present invention described previously,the values stored in the cells of the input layer and the values storedin the cells of the memory layer are held constant. To perform selectiveattention the cells which represent the background portion of the imageor pattern stored in the memory layer are allowed to converge to anyvalue as required.

In performing Selective Attention the background pixels of an image thatare stored in the LTM memory layer are “turned loose.” They can convergeto any value. This requires that the Convolution that is used in theHierarchical Zooming process that is part of the Ebbing Resonance mustbe redefined.

Referring to FIG. 6, if the two input/STM and memory/LTM patterns are atwo dimensional encoding of the edges of an image. An image can beencoded on a pixel board using a physical phenomena such as electricfields. If two additional layers are added, one prior to the input layerand one after the memory layer, then the distance between the pre-inputlayer and the input/STM layer serves for “blurring the image.” Arelaxation can be achieved by gradually decreasing the distance betweenthe pre-input and the input layer. The same is done with the memorylayer. The distance between the memory layer and the post-memory layeris gradually decreased, the result is that both the input and the memorypatterns gradually become clearer. Now the pre-input and post memory/LTMlayers are held constant instead of the original input/STM andmemory/LTM layers. The desired behavior of the system can bemathematically shown, the electric field potential depending on I/Rwhere R is the distance from the electric charge. Thus, as the distancer between the memory/LTM and the post memory/LTM board grows, thepotential difference between two points on the post memory/LTM board asseen on the memory/LTM board as,

${{Potential}\mspace{14mu}{difference}} = {\frac{q_{1}}{\sqrt{r^{2} + d_{1}^{2}}} - \frac{q_{2}}{\sqrt{r^{2} + d_{2}^{2}}}}$such that q₁ and q₂ are the charges of pixel₁ and pixel₂ respectivelyand d₁ and d₂ are the “horizontal” distance from the interacting pointon the memory/LTM board. As r increases, the influence of d₁ and d₂ isreduced and the image on the memory board becomes blurred. The loosecells in this case are on the post memory layer.

Note that while the examples used to explain the invention used a matrixcontaining 6 layers with each layer including eight cells or units, thepresent invention may include a matrix with any number of layers witheach layer containing any number of cells. As the number of layersincreases the granularity of the process is reduced such that differencein values stored from one cell to an adjacent cell is minimized.

Note also that while the examples used to describe the present inventionwere limited to a one-dimensional numeric transformation engine, otherembodiments exist, including multi-dimensional matrices and data spacesthat are within the present invention.

The embodiments of the present invention described thus far incorporatea digital means to produce the metamorphosis of the patterns from theLTM pattern to the STM pattern. Instead of using digital numericcalculations another embodiment of the present invention uses electricfields to accomplish the time varying convolutions and the vibrationwaves used to complete the gradual metamorphosis. In this physicalembodiment, the units that constitute the layers between the input/STMlayer and the memory/LTM layer may be either passive (for exampleinduced or constant dipoles) or active (for example CMOS components). Ineither case, the units that constitute the intermediate layers cannot befixed in space. If passive units constitute the intermediate layers,these passive units cannot be spherical in shape, but must provide someform of asymmetry with respect to a desired physical characteristic,e.g., electric change (dipole), magnetic orientation (North/South); etc.One requirement of the passive units is that they be able to point inany direction, i.e., free to orient in a position under influence of afield.

The resonance and the update waves, the “reduction of the intensity ofthe waves” (The Simulated Annealing) and the Far Span k-Distanceconnections (Far Differentials) described thus far employ a numeric,digitally implemented means of achieving a gradual metamorphosis betweenthe input STM pattern and the memory LTM pattern such that in everyintermediate pattern the Topology (Shape) is preserved. That is,manipulation of data patters products a numeric representation of themetamorphasic.

In the digital means of achieving a transformation engine and otherstrongly cooperative systems, local coupling of degrees of freedom leadto scaling phenomena through a cascade effect which propagatesthroughout the entire system. The transformation engine approachgenerates new potentials (known as Hamiltonians) from older potentialsby incrementally removing degrees of freedom during each iteration. Atthe completion of each iteration a Hamiltonian representing unchangedlength scale interaction remains.

The present invention also includes a physical transformation enginewhich includes mixed terms in corresponding Hamiltonian. In this case,it is unnecessary to progressively integrate-out degrees of freedom bythe use of Ebbing Resonance and For Spin K-distance connection.

FIG. 7 shows a dimensional hexagonal implementation of a transformationengine according to an embodiment of the invention. A cell or unit in atwo dimensional numeric transformation engine can be generallyrepresented by Unit (s, t, w) where “s” represents the layer and “t, w”represent the coordinates of the unit within the layer. In thetwo-dimensional transformation engine unit (s, t, w) now interacts withseven units in layer s+1 and seven unit in layer s−1. Unit (s, t, w)“examines” (i.e., the following nearby units are examined in formulatingan updated value to store into unit (s,t,w)) unit (s+1, t+1, w+1), unit(s+1, t+1, w−1), unit (s+1, t−1, w+1), unit (s+1, t−1, w−1), unit (s+1,t, w+2), unit (s+1, t, w−2) and unit (s+1, t, w). Unit (s, t, w) alsointeracts with seven units in layer s−1 namely: unit (s−1, t+1, w+1),unit (s−1, t+1, w−1), unit (s−1, t−1, w+1), unit (s−1, t−1, w−1), unit(s−1, t, w+2), unit (s−1, t, w−2) and unit (s−1, t, w).

The transformation engine may be described by using the idea ofMinimization of an Action Integral and Calculus of Variations in orderto describe a potential p (scalar function p(x, y, z, t)) that preservesthe geometric properties of itself as it changes from the input STMpattern to the memory LTM pattern. Note that a single potential isinsufficient in order to generate realworld metamorphosis betweenpatterns. Also note that a physical transformation engine should work ontwo-dimensional patterns of Grey Scale patterns.

The minimization of an Action Integral may be used to derive Einstein'sGravity equations and the Quantum Mechanics equations. As furtherdescribed below, the transformation engine may be described in terms ofan Action Integral. Additionally, the Action Integral of thetransformation engine may be written in Tensor form and may haveapplications in String Theory.

Both Far Span k-Distance Connections and the Ebbing Resonance processare redundant in a physical implementation of a transformation engineconsistent with the invention. A hardware transformation engine iscapable of performance beyond the Turing limit, meaning that a puresoftware solution will not be able to perform as well as thetransformation engine. Instead, a physical hardware based transformationengine demonstrates a behavior that is dictated by a local Hamiltonian.For this reason the Far Span k-distance connections are renderedredundant. Far and near geometric features are encoded by SpatialFrequency functions. As will be shown the principles ofDielectrophoresis and of Electrorotation are an outcome of theHamiltonian that is minimized by the transformation engine.

An Action Integral is an integral over space and time of an operatorover a function. The operator in the integral is a Lagrangian. If theintegral is a sum of a potential and the kinetic energy then theLagrangian is an Hamiltonian.

Considering the classical physics action of a charged particle in anelectric field of another charge that is fixed. The Hamiltonian (L) forspeed (v) mass (m) and positive (q1) and negative (q2) charges, and onefree particle coordinates x(t), y(t), z(t) such that time (t) is givenas:

$L = {{{- \frac{q_{1}q_{2}}{r^{2}}} + {\frac{1}{2}{mv}^{2}}} = {{- \frac{q_{1}q_{2}}{x^{2} + y^{2} + z^{2}}} + {\frac{1}{2}{m( {{( \frac{\mathbb{d}x}{\mathbb{d}t} )^{2} + ( \frac{\mathbb{d}y}{\mathbb{d}t} )^{2} + {( \frac{\mathbb{d}z}{\mathbb{d}t} )^{2}{{If}\mspace{14mu}{Action}\mspace{14mu}{over}\mspace{14mu}{space}\mspace{14mu}{time}\mspace{14mu}{is}\mspace{14mu}{minimized}}}} = {{{Min}\mspace{14mu}{\int_{t}{\int_{x}{\int_{y}{\int_{z}{{L \cdot \ {\mathbb{d}x} \cdot \ {\mathbb{d}y} \cdot \ {\mathbb{d}z} \cdot \ {\mathbb{d}t}}\mspace{14mu}{then}\mspace{14mu}{the}\mspace{14mu}{{action}'}s\mspace{14mu}{variation}\mspace{14mu}{vanishes}0}}}}}} = {\delta\;{\int_{t}{\int_{x}{\int_{y}{\int_{z}{L \cdot \ {\mathbb{d}x} \cdot \ {\mathbb{d}y} \cdot \ {\mathbb{d}z} \cdot \ {{\mathbb{d}t}.}}}}}}}}}\mspace{14mu} }}}}$

Note that the Hamiltonian of the transformation engine will have ahardware implementation using units that behave like induced electricdipoles in the sense that they depend on the gradient of the localelectric field E.

The Hamiltonian is an operator that is defined over the location vectorfunction (x(t),y(t),z(t)). Minimizing the Action Integral results inmotion equations of the particle that are not fixed. In the calculus ofvariations the minimization of an action integral simply means that thevariation of the action Hamiltonian is either 0 or vanishes.

The transformation engine provides: 1) intermediate patternsrepresenting a metamorphosis between and affected by both input STMpattern and memory LTM pattern, and 2) intermediate patterns that changebetween the STM and LTM layers such that their respective geometriccharacteristics are not lost.

FIG. 8 is an illustration of either an input STM or LTM including pixelsof metal 801, where each pixel of metal 801 is fed with a differentalternating or constant voltage at different frequencies. This physicalimplementation of a transformation engine model provides a solution inwhich pixels of the same layer/board will not interact with each otherin a way that will disrupt the generation of the plurality ofmetamorphosis patterns between the input STM pattern and memory LTMpattern. From the metamorphic between the LTM and STM patterns and thepreservation of geometric characteristics, the transformation engineHamiltonian is the sum of at least two terms. One term forces theintermediate values of the potential field, that represents a changingpattern, to change between the STM and the LTM patterns and the secondterm is responsible that geometric information will not be lost inbetween. A tensor form transformation engine Hamiltonian will also bepresented causing a minimization of energy and minimization of the lossof geometric information. Note that, in the physical transformationengine, the action integral need not depend on time although a “full”physical solution must depend on time.

In a two dimensional transformation engine a gradual metamorphosis isgenerated between the input STM pattern and a memory LTM pattern. If thepattern is encoded by potentials (or some other scalar function) theneach pixel in the STM and LTM boards will hold a different potential.For the intermediate potential between the two boards to graduallychange it is sufficient to require that the integral of the square ofthe gradient of the potential to be minimized. In electric field terms,that fits the minimization of the integral on the squared electricfield. Multiplying this value by the half of the permitivity factoryields the static electric field energy,

${Energy} = {\frac{ɛ_{0}}{2}{\int{\int{\int( {( \frac{\mathbb{d}p}{\mathbb{d}x} )^{2} + ( \frac{\mathbb{d}p}{\mathbb{d}y} )^{2} + {( \frac{\mathbb{d}p}{\mathbb{d}z} )^{2}{{\mathbb{d}x} \cdot {\mathbb{d}y} \cdot {\mathbb{d}z}}}} }}}}$

The incomplete Hamiltonian of a transformation engine may be written as:

$L = ( {{( \frac{\mathbb{d}p}{\mathbb{d}x} )^{2} + ( \frac{\mathbb{d}p}{\mathbb{d}y} )^{2} + ( \frac{\mathbb{d}p}{\mathbb{d}z} )^{2} + {{Unknown}\mspace{14mu}{term}}} = {{{{\nabla p} \cdot {\nabla p}} + {{Unknown}\mspace{14mu}{term}\mspace{11mu} 0}} = {\delta{\int{\int{\int_{Space}{( {L + {{Unknown}\mspace{14mu}{term}}} )\ {{\mathbb{d}x} \cdot {\mathbb{d}y} \cdot {\mathbb{d}z}}}}}}}}} $

0 in the expression above means that the variation of the integralvanishes. This Hamiltonian is equivalent to the square distance betweenunits previously described with respect to reaching the ebbing resonancestage and using the for span k-distance connection to amend theEuclidean distance. In a transformation engine that explicitly usesplurality of layers, information is not lost as a pattern changes fromone intermediate layer to an adjacent intermediate layer. As previouslydescribed, the Far Span k-Distance Connections is responsible for theconservation of geometry.

Writing this term locally means that the difference of a potential p1and p2 on layer S will not change excessively in view of thecounterparts q1, q2 in layer S+1 or S−1. Thus, ((p₂−p₁)−(q₂−q₁₎₎ ² mustbe added to the Hamiltonian, wherein the direction p1 to q1 is thedirection of the derivative of the potential.

The difference between two adjacent intermediate potential layers isalso held constant. Referring to FIG. 9, if the Z axis is the directionin which the potential changes and the adjacent potentials along the Zaxis are p1 (901) in layer S, q1 (902) in layer S+1 and w1 903 in layerS+2 then it is desirable to minimize ((p1−q1)−(q1−w1))². In general anobjective is to minimize ((p₂−p₁)−(q₂−q₁))²+((p1−q1)−(q1−w1))² along thetransformation direction. For ease of explanation, Z will be as thelocal direction of the gradient P. The sum of the last two terms asdifferentials is then equal to

$\frac{( {( {{p\; 2} - {p\; 1}} ) - ( {{q\; 2} - {q\; 1}} )} )^{2} + ( {( {{p\; 1} - {q\; 1}} ) - ( {{q\; 1} - {w\; 1}} )} )^{2}}{({dZ})^{2}}$

In order to write the last term such that the potential p may change inany direction, the general form is (∇∇P·∇P)·(∇∇P·∇P).

In second derivative terms this term can be written as,

-   -   Term₂ in the lagrangian is the scalar (∇∇P·∇P)·(∇∇P·∇P)

${\nabla P} = {{\begin{pmatrix}\frac{\partial p}{\partial x} \\\frac{\partial p}{\partial y} \\\frac{\partial p}{\partial z}\end{pmatrix}{\nabla{\nabla P}}} = \begin{pmatrix}\frac{\partial^{2}p}{{\partial x}{\partial x}} & \frac{\partial^{2}p}{{\partial x}{\partial y}} & \frac{\partial^{2}p}{{\partial x}{\partial z}} \\\frac{\partial^{2}p}{{\partial y}{\partial x}} & \frac{\partial^{2}p}{{\partial y}{\partial y}} & \frac{\partial^{2}p}{{\partial y}{\partial z}} \\\frac{\partial^{2}p}{{\partial z}{\partial x}} & \frac{\partial^{2}p}{{\partial z}{\partial y}} & \frac{\partial^{2}p}{{\partial z}{\partial y}}\end{pmatrix}}$

The second derivative matrix is multiplied by a vector so

$( {{\nabla{\nabla P}} \cdot {\nabla P}} ) = {{\begin{pmatrix}\frac{\partial^{2}p}{{\partial x}{\partial x}} & \frac{\partial^{2}p}{{\partial x}{\partial y}} & \frac{\partial^{2}p}{{\partial x}{\partial z}} \\\frac{\partial^{2}p}{{\partial y}{\partial x}} & \frac{\partial^{2}p}{{\partial y}{\partial y}} & \frac{\partial^{2}p}{{\partial y}{\partial z}} \\\frac{\partial^{2}p}{{\partial z}{\partial x}} & \frac{\partial^{2}p}{{\partial z}{\partial y}} & \frac{\partial^{2}p}{{\partial z}{\partial y}}\end{pmatrix}\begin{pmatrix}\frac{\partial p}{\partial x} \\\frac{\partial p}{\partial y} \\\frac{\partial p}{\partial z}\end{pmatrix}} = \begin{pmatrix}{{\frac{\partial^{2}p}{{\partial x}{\partial x}}\frac{\partial p}{\partial x}} +} & {{\frac{\partial^{2}p}{{\partial x}{\partial y}}\frac{\partial p}{\partial y}} +} & {\frac{\partial^{2}p}{{\partial x}{\partial z}}\frac{\partial p}{\partial z}} \\{{\frac{\partial^{2}p}{{\partial y}{\partial x}}\frac{\partial p}{\partial x}} +} & {{\frac{\partial^{2}p}{{\partial y}{\partial y}}\frac{\partial p}{\partial y}} +} & {\frac{\partial^{2}p}{{\partial y}{\partial z}}\frac{\partial p}{\partial z}} \\{{\frac{\partial^{2}p}{{\partial z}{\partial x}}\frac{\partial p}{\partial x}} +} & {{\frac{\partial^{2}p}{{\partial z}{\partial y}}\frac{\partial p}{\partial y}} +} & {\frac{\partial^{2}p}{{\partial z}{\partial y}}\frac{\partial p}{\partial z}}\end{pmatrix}}$

Adding the energy Hamiltonian to the Hamiltonian that preserves thegeometry in local terms yields:

-   -   Hamiltonian=∇P·∇P+K(∇∇P·∇P)·(∇∇P·∇P)+Optional additional terms        where K is some constant.

The optional terms may involve higher derivatives of the potential P.This function is the Hamiltonian of a transformation engine according tothe present invention; the plurality of metamorphosis patterns betweentwo boundary conditions which are the input STM pattern and the memoryLTM pattern as described.

Note that a second derivative of the potential is multiplied by thefirst derivative resulting a product in the form of a vector. Such aterm is consistent with elongated induced dipoles in a local electricfield. This is because electric dipoles respond to or “feel” force whenthey are positioned in an electric field such that the second derivativedoes not vanish. Elongated dipoles that are made of conductive materialalign with the first derivative of the potential and therefore theHamiltonian that is the result of the previous illustration implicitlydictates the use of elongated dipoles. In order to be totally consistentwith the Hamiltonian the dipoles should be ideal needles at sufficientlyhigh resolution and with no or minimal resistance.

Dielectrophoresis itself is insufficient to explore the description ofelongated induced dipoles. The net force on an isotropic elongateddipole in an electric field depends on the fields alternating andconstant components e.g., V=V_(base)+A*cos(ωt) such that ω is thefrequency of the alternating component, A is the amplitude in voltageand V_(base) is the voltage baseline. The force on a spherical dipolemay be expressed as:F _(Dipole)=2πR ³ε_(m) Re[k(ω)]∇E ²

Such that K is the Clausius Mossotti constant. This refers to therelationship:

${k(\omega)} = \frac{ɛ_{p}^{*} - ɛ_{m}^{*}}{ɛ_{p}^{*} + {2ɛ_{m}^{*}}}$such that

$ɛ^{*} = {ɛ - \frac{j\sigma}{\omega}}$and σ is the conductivity of the particle

wherein ‘m’ and ‘p’ stand for medium and particle (respectively) and jis the square root of −1. Note that the third power of R makes itdifficult to exert strong forces using spherical particles especiallywhen R is very small. Further, the conductivity is extremely importantin order to achieve highly responsive particles. A more complex model isof particles on which the electric force depends on the frequency ω. Ahighly conductive needle shaped particle redirects the electric fieldbecause the potential depends on the orientation of the needle. The term2πR³ can be ignored for needles. Thus, the needle will be parallel tothe field. We know from the simple case of constant dipoles that theminimum energy of a dipole D in a field E is u=−D*E where D is thedipole.

The transformation engine Hamiltonian and the square force on a localdipole are linearly dependent

$( {\frac{1}{2}( {\nabla E^{2}} )} )^{2} = {{( {E \cdot {\nabla E}} ) \cdot ( {E \cdot {\nabla E}} )} \propto {F_{Dipole}^{2}.}}$

Note that minimizing the geometric part of the transformation engineHamiltonian means that the integral of the square force on all thedipoles is minimized. This result may not be readily apparent in lightof the energy of a spring is

${\frac{1}{2}{KX}^{2}} = {{\frac{1}{2K}( {K^{2}X^{2}} )} = {\frac{1}{2K}F^{2}}}$where K is the spring constant.

The variation of the transformation engine Hamiltonian must vanishbetween the input STM pattern (first boundary condition) and the memoryLTM pattern (second boundary condition), so:

$0 = {\delta\underset{x,y,z}{\int{\int\int}}( {{{\nabla P} \cdot {\nabla P}} + {{K( {{\nabla{\nabla P}} \cdot {\nabla P}} )} \cdot ( {{\nabla{\nabla P}} \cdot {\nabla P}} )}} ){{\mathbb{d}x} \cdot {\mathbb{d}y} \cdot {\mathbb{d}z}}}$

If the number of layers in the transformation engine is sufficientlylarge and the distance between the STM and LTM is also sufficientlylarge and fields are relatively strong then the constant K can be verylarge. In practice the transformation engine can work with very highterms of K of about 4096 and if the gradient of p is calculated over farpoints within the same layer then K can even be 10⁵ or larger. Thelarger the K constant is, the closer the transformation engine is to acomplex springs machine. Note that for practical applications of thetransformation engine one potential is insufficient.

The theory about spherical isotropic particles that are placed in anelectric field is insufficient when the shape of the particle is notspherical. The Hamiltonian of the transformation engine requires idealneedle shaped particles. The dependence of the net force on a dipoledepends of the gradient of the electric field and that feature isapplicable also in non-spherical dipoles.

Instead of using a single potential, p, several potentials should beused in order to resolve two dimensional ambiguities. That is, e.g.,different alternating voltage frequencies and elongated pieces ofmaterial/particles/proteins/neurotransmitters that reside between theLTM and STM pixel boards can serve as vector fields because an elongatedparticle points to a direction and behaves like a needle that is placedin electric field and turns until it reaches equilibrium in minimumenergy. If such “needles” respond to different frequencies and each“needle” is wrapped in an isolating ball, then the needles align likevectors in local fields. Since needle characteristics may be selected sothat the needles selectively respond to different frequencies, thesystem of two pixel boards, STM and LTM and dipoles that respond todifferent voltage frequencies—“different needles”—simulate a multipotential system.

The Hamiltonian of multiple potentials P(i) may be expressed as:

${\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{{\nabla{P(i)}} \cdot {\nabla{P(j)}}}}} + {{K( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )} \cdot ( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )}$such that the variation over the n potentials vanishes, as follows:

$0 = {\delta\underset{x,y,z}{\int{\int\int}}{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{( {{{\nabla{P(i)}} \cdot {\nabla{P(j)}}} + {{K( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )} \cdot ( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )}} ){{\mathbb{d}x} \cdot {\mathbb{d}y} \cdot {\mathbb{d}z}}}}}}$in a gray scale image. This mathematical formulation is consistent withat least one preferred embodiment of a transformation engine accordingto the invention.

The mixed term of:

$\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}( {{{\nabla{P(i)}} \cdot {\nabla{P(j)}}} + {{K( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )} \cdot ( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )}} )}$is made of two parts, namely:

The part of the Hamiltonian that forces the pattern to gradually changefrom the input STM to the memory LTM pattern may be expressed as:

$\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{{{\nabla{P(i)}} \cdot {\nabla P}}(j)}}$

Minimization of this term alone means an electric like field linesbetween the input STM pattern and the memory LTM pattern:

-   -   a) The part of the Hamiltonian that forces the pattern to        maintain its geometric properties as it gradually changes from        the input STM pattern to the memory LTM pattern is

$ {K*{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}}}}} ) \cdot {\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}$such that K is some constant.

A gray scale image can be decomposed into high and low frequency FourierTransforms. A single picture can be translated into multiple picturesusing either Fourier Transforms or Wavelet Transforms. In an input STMpattern there are wide vertical stripes and an LTM pattern where thestripes are narrow, the Fourier Transform frequencies that will maximizeon the STM pattern will be lower than the Fourier Transform frequenciesthat will maximize on the input STM pattern. The Hamiltonian that isminimized therefore must include mixed terms if the potentials P(i) needto represent different frequencies Fourier Transforms.

A Hamiltonian that is made of different potentials reduces the degreesof freedom of the transformation between the input STM pattern and thememory LTM pattern.

A potential is a function from R^(n) to R. The explicit second order(second derivatives) Hamiltonian is:

$\sum\limits_{i = 1}^{n}\;{\sum\limits_{j = 1}^{n}\;( {{{\nabla{P(i)}} \cdot {\nabla{P(j)}}} + {{K( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )} \cdot ( {{\nabla{\nabla{P(i)}}} \cdot {\nabla{P(j)}}} )}} )}$such that P(i) are the potentials.

Renormalization group methods and introduced to provide a method oftreating systems in which multiple degrees of freedom spanning manyscales of length are locally coupled. At the end of each step aHamiltonian results that represent the interactions over length scalesnot yet treated. In particular, changes in gray-scale levels occur overmany scales of length for which the image is organized. Thus, thetransformation engine integrates many prior data that comes from lowerlevel neurons such as V1, V2, LGN and integrates all this data intosingle high-level Neurons.

Coupling, is in terms of the Hamiltonian because the different scales offeatures in one image may fit different scales of features in another.For example wide lines in the input image may be fit to narrow lines inthe memory image. The fitting problem may be even more complex. As shownin FIG. 10, it may be desirable to map a disk that is full of + signs1001 to a disk full of − signs 1002 or to map such rectangles withoutcoupling. Coupling is a mathematical term that is a part of theHamiltonian of a system such that it includes multiplication offunctions with different indices. (E.g.,f₁(x,y)*f₂(x,y)+f₁(x,y)*f₃(x,y)+f₂(x,y)*f₃(x,y) that is a sum of threecoupled terms f₁(x,y)*f₂(x,y), f₁(x,y)*f₃(x,y) and f₂(x,y)*f₃(x,y) suchthat x,y are planar coordinates.)

Coupling may involve dipoles that are subject to maximum torque forcewhen exposed to two different frequencies of alternating voltage. Thusthe mixed terms in the transformation engine Hamiltonian can beimplemented using, for example, Evotec OAI technology. A parceling oftwo particles that each responds to different frequency provides anexact solution to the coupling problem.

Two boundary conditions of the LTM and STM are well defined. Those arethe LTM board and the STM board. The potential LTM=P(X,Y,0) andSTM=P(X,Y,1) denote the LTM and STM patterns respectively. Theconvergence of the mapping between the LTM and the STM such that thetopology is preserved depends on the distance between the LTM and theSTM. Based on this constraint, the gradient of the potential should beas perpendicular as possible to the LTM and STM boards.

Referring to FIG. 11, the “+” sign 1101 is encode by a 5000 Hzalternating voltage that is fed to part of pixels in memory pattern and1000 Hz encode the ‘−’ sign 1102 on the input pattern. Units thatrespond to both 5000 Hz and to 1000 Hz generate the map between theinput pattern and the memory pattern.

The transformation engine Hamiltonian of a single potential isconsistent with the laws defined by General Relativity. It is probablethat the description of matter in the cosmos as a transformation enginerequires more than one potential. However, for the purposes of thepresent illustration, one potential is used herein and developed intotensor form for simplicity of explanation. The transformation engine cantake an interesting General Relativistic form with the followinggeometric implications. To express the transformation engine in terms ofcovariance principle articulated by Einstein, it is sufficient toreplace the x,y,z parameterization with x,y,x,ct such that c is thespeed of light and t is the time, and to replace each derivative with aCovariant or with a Contravariant derivative. Einstein's summationconventions will be used (upper and lower indices are summed up).

Writing L in Einstein's tensor convention provides:L=(p ^(i) p _(i) +Kp ^(n) p _(k);_(n) p ^(m) p ^(k);_(m))√−gsuch that p is the pattern potential that gradually changes between theSTM and the LTM patterns.

In the present case, LTM can be a known distribution of the potentialfield p in the past t=0 and the STM can be a current distribution fort=1 such that a description of the cosmos may be expressed in terms ofwhat happens in-between. In such a case the Hamiltonian L can have termshigher orders for example:L=(p ^(i) p _(i) +Kp ^(n) p _(k);_(n) p ^(m) p ^(k);_(m) +K ^(4/3) p^(n) p _(k);_(L);_(n) p ^(m) p ^(k);^(L);_(m))√−gwhere −g is the minus sign of the determinant of the metric tensor, thesquare root of −g is the scaling factor of a volume element, where K isa constant and the semi colon; denotes the covariant derivative. Thepower 4/3 is provided to account for dimensionality. The Hamiltonian canbe written in another formL=(p ^(i) p _(i) +p ^(n) A _(kn) p ^(m) A ^(k) _(m) +p ^(n) B _(kLn) p^(m) B ^(kL) _(m))√−gwhere A and B are high-order tensors

-   -   if p^(m)p^(k);_(m)=0 then P^(m) is geodetic.

In terms of differential geometry p^(m)p^(k);_(m) is an evaluation ofhow parallel the field p^(m) changes in relation to itself. If this termis 0 then p^(m) is a geodetic field and it represents an inertialparticle. If this term is not 0 then it means that a “force” acts onmatter. For this reason the transformation engine can be one of themodels a unified force theory of the fine fundamental forces.

Using the Christoffel symbols {kn,u}, comma for derivative and g_(ij) asthe metric tensor, results in the relationship:L=(P, _(m) P, _(k) g ^(mk) +K g ^(LK)(p, _(j) g ^(in)(P, _(k),_(n)−{kn,u}P, _(u))(P, _(j) g ^(in)(P, _(L),_(n) −{Ln,u}P, _(u)).)√−gthe square root of the −g term is always used in General Relativity forthe measurement of a 4-volume element on the space-time manifold.

The Cristoffel symbols written explicitly provides:

$\{ {{kn},u} \} = {\frac{1}{2}{g\;}^{\lambda\; u}( {g_{k\;\lambda},_{n}{+ g_{n\;\lambda}},_{k}{- g_{kn}},_{\lambda}} )}$

The variation over the space-time manifold should vanish, that is,

$0 = {\delta{\int_{\Omega}( {P,_{m}P,_{k}{g^{mk} + {{Kg}^{Lk}( {P,_{j}{{g^{jn}( {P,_{k},_{n}{{- \begin{Bmatrix}u \\{kn}\end{Bmatrix}}P},_{u}} )}( {P,_{j}{{g^{jn}( {P,_{L},_{n}{{- \begin{Bmatrix}u \\{Ln}\end{Bmatrix}}P},_{u}} )}\sqrt{- g}\mspace{7mu}{\mathbb{d}\Omega}}} }} }}} }}$such that Ω is the 4-volume space-time domain.

Transformation engines according to another embodiment of the presentinvention may also be constructed using FM and microwave frequencyelectric diploes. As described, the transformation engine uses unitsthat align and generate a plurality of intermediate patterns between theinput STM pixels board and the LTM memory board. One method ofaccomplishing this is by using a spring or a coiled shaped polymer thatis coated with metal. The two tips of the spring are ball shaped and theentire spring is embedded in a spherical polymer as appears in FIG. 12.

Each unit in FIG. 12 includes a sphere 1201. Within sphere 1201 there isa liquid with a dielectric constant (K_(e)) 1202 as close to 1 as ispossible with some types of oils. The core 1203 of sphere 1201 is freeto turn sphere 1201 (such as a baseball shaped polymer). Core 1203 may,for example, contain coil 1204 with two ball shaped tips 1205 and 1206attached. For proper functioning of the transformation engine the unitsmust be free to be aligned within a field. For example, the dipole maybe baseball shaped, spring shaped or in the shape of a needle. Sphericaldiploes do not redirect the external field and can only performconformal maps because of Laplace' equation. Conformal maps map tangentcurves to tangent curves and thus spherical dipoles (that effectivelychange the permitivity of space) should generally not be employed by thetransformation engine.

The approximated maximum force that will be felt by the unit depends on1/LC where that L is the coil's magnetic induction constant in units ofHenries and C is the capacitance of the dipole. A formulation of theoptimum frequency of an external electric field also depends on otherfactors such as resistance.

A single unit can contain two different parallel coils and can thusrespond to two independent frequencies. This property is useful in orderto obtain appropriate coupling. Coupling in the transformation enginesense is broader than in Ising spin models because real induced dipolescan point to any direction and the coupling relates to frequencies andnot of quantum spins. The coil or spring shaped dipole or two dipoles ineach unit cause the unit to turn, or align, in response to local fieldsat different frequencies (e.g., 10000 KHz and 57000 KHz) and other wholemultiplicity of these two basic frequencies (e.g., n*10000 KHz andm*57000 KHz) wherein m and n are whole numbers greater than or equal to1.

These units can be placed on many boards, or inner layers, between theinput pixels board (containing the STM and LTM patterns) and they can besuspended in liquid between the input and the memory boards.

As shown by FIG. 13, the inner layers, in solid systems, may beconsidered boards 1301 of spherical cells in which these units 1302 aresuspended. The number of possible solid configurations is limited by thecorresponding physics.

As shown in FIG. 14, one example solution is to mix many different unitsof 40 nm of size between the input and memory boards. These units arethen suspended in water or other suitable liquid or, more generally,fluid and respond to the electrodes that are connected to the middle ofeach Input and memory pixel. This configuration may be viewed as a“liquid transformation engine”.

Camera 1401 is used to convert an image into electrical signals on pixelboard 1402. Edge detectors check groups of adjacent pixels forhorizontal, vertical and diagonal lines. Identified edges are containedin pixel board 1403. This may be accomplished by the edge detector whichidentifies the edge firing at a certain frequency into pixel board 1403.Additional and more complexed detection may also be performed on thepixel boards. For example, a object recognition software may be used toidentify various objects such as a human heel, an ear, a nose, or otherpredefined objects. Detection units which objects within portions of thepixel image may also fire at specific frequencies into correspondingpixels of pixel board 1404. Both pixels boards 1403 and 1404 areconnected to transformation engine 1405. Specific pixels in thetransformation engine's input pixel board receives pulses both for theexistence of an edge and for the existence of identified objects.Electric dipoles 1406 within the transformation engine align such thatthe energy between the input board 1407 and the memory board 1408 isminimized. Within the transformation engine the suspended units maneuveror are otherwise positioned and/or oriented to accomplish this alignmentwithin the surrounding material, such as the liquid. Individual unitsmay respond to different frequencies and create a mapping betweendissimilar objects such as an input of “x's” to a memory of “o's”. Thememory board is connected to a regeneration oscillator chip. The chip isable to “learn” (i.e., characterize) the pulses coming from the inputboard and later to regenerate these pulses. Object recognition unit 1409analyzes the signals received from the input board. If these signalsmatch the learned and stored signals then the transformation enginefires in order to indicate or signal a match between the image and thestored information.

Note that a solid solution may be less efficient than a liquid orgaseous solution. In the solid machine the spherical units also have acore that is an elongated dipole, possibly spring shaped, that is freeto turn in space. The units should have a limited angular freedom ofmovement. The spheres denote the cells that contain spring shapedelongated dipoles.

FIGS. 15A and B illustrate a solid structure which may be used as atransformation engine. In this embodiment the dipolar units in a dipolartransformatron can be ordered in a cyclic structure of three planarlayers. In each such layer the elongated dipoles are positioned in thevertices of 60°, 60°, 60° triangles. Each layer is shifted downward by1/√3 in relation to its previous layer. Due to differences in the numberof units along diagonal paths and perpendicular paths from the input tothe memory layer, and vice versa, the space between the layers (theinterlayer substance) is filled with non homogenous material; otherwisethe layer's structure is totally homogenous.

FIG. 15A illustrates layers S, S+1, S+2 on top of each other while FIG.15B illustrates top layer S 1501. Referring to FIG. 15A, layers S(1501), S+1 (1502), S+2 (1503) show a cyclic structure of three layers.The structure of each layer is just 60°, 60°, 60° isosceles trianglesthat optimally cover or tile the plane. The layers are identical instructure. Each two adjacent layers are shifted by a distance of 1/√3units of length. Each of the triangle sides is 1 unit long. Thegeometric intersection of the medians and the heights of each trianglein layer S is the same. The distance of each vertex from the heightsintersection is

$\frac{\frac{1}{2}}{y} = {{\cos( {30{^\circ}} )} = { \frac{\sqrt{3}}{2}\Leftrightarrow y  = \frac{1}{\sqrt{3}}}}$as in FIG. 16.

FIG. 16 shows the orientation of the layers when the distance of eachvertex from the heights intersection is

$\frac{1}{\sqrt{3}}.$In order to allow stationary transformations with this embodiment, eachunit in layer S should have the same distance from its counterpart inlayer S+3 as from the closest 3 units in layer S+1.

Note that some energy will be required for a dipole to change positionbecause of the elongated nature of the dipole. This elongation alongwith the nonlinear nature of dipole-dipole interactions allows localcompetition, a trait important for topology conservation and fundamentalto a transformation engine according to a present embodiment of theinvention.

Each unit in layer S must have the same distance from four other units,one in layer S+3 and 3 in layer S+1. This condition imposes thefollowing distance d between layers,

${3\; d} = \sqrt{( \frac{1}{\sqrt{3}} )^{2} + {d^{2}.}}$The distance between layer S to S+1 is d and therefore the distance fromthe counterpart dipole in layer S+3 is 3d. The distance to each one ofthe closest units in layer S+1 is by Pythagoras

$\sqrt{( \frac{1}{\sqrt{3}} )^{2} + d^{2}}.$From this equation

$d = {\frac{1}{\sqrt{24}} \cong 0.204}$and the distance between the closest units is therefore R=3d≅0.6124.

An important value is the ratio

$Q = \frac{1}{R^{4}}$that is an approximated ratio between the forces of closest units withinlayer S and between each unit in S and each one of the 4 closest units,3 in layer S+1 and one in layer S+3.

Q=7 1/9. This high quotient implies that most of the influence on eachdipole comes from interactions between adjacent layers.

Another issue addressed by the transformation engine resolution. Inparticular, if the transformation is stationary then the layersresolution is only ⅓ in the case where the transformation includes allthe layers. This fact implies an emergent property of the triangularstructure of the dipolar transformatron; different levels oftransformations use different paths. A collision between two paths maybe a sign that there is a contradiction in the mapping process andtherefore an AC signal, even if the induced dipoles are quick, will leadto a welcome instability.

As mentioned previously, preprocessing may be performed on the valuesassociated with the input pattern and the memory pattern. For example,preprocessing may be performed to replace the ACCGTGGA sequence for DNAcomparison with a logical “1”. The complement of the ACCGTGGA sequenceis a CCCTGGCACCTA sequence because the ACCGTGGA sequence may bind to theCCCTGGCA sub-string. Preprocessing may be performed on the CCCTGGCACCTAsequence to replace the sequence with a logical “0”. The preprocessedvalues may then be inserted into the appropriate layers.

Another embodiment of the invention provides a Transformatron model inRiemannian Geometry. In contrast to the prior embodiment implementing aFar Span K−Distance Connections, the following embodiment uses covariantgeneralization in Riemannian geometry and thus allows complex matchingbetween low dimensional patterns which is more difficult in Euclideangeometry.

As was described above in connection with the first embodiment, atransformation between boundaries, memory layer and input layer (e.g.ellipse drawn on the plane x,y,0 in R³ as p(x_(ellipse),y_(ellipse),0)=1and 0 is elsewhere in the plane where the ellipse curve is not presentand between a circle drawn on the plane x,y,1) can be done such that anintermediate pattern on the intermediate plane x,y,0.5 will emerge (e.g.an ellipse that is an intermediate pattern between the ellipse on theplane x,y,0 and between a circle on the plane x,y,1 in R³). This is doneby defining a local cost function such that minimizing that local costfunction causes a potential function p to be defined throughout theentire region between the boundaries. This method is used in order tomatch patterns by a later step that was named, “vibration waves”.

By matching through curved space, more complex matching tasks can besuccessfully achieved as curved geometry allows parallel lines to existbetween the input pattern and the memory pattern even if in flatgeometry this task is impossible. For example one can imagine parallelstraight lines connecting a circle on x,y,0 plane with a circle drawn onthe x,y,1 plane. If instead of circle, an eccentric ellipse is drawn onthe x,y,0 plane then one can't match between the ellipse on x,y,0 and acircle on x,y,1 by using only straight parallel lines. This task,however, can be achieved if the space itself is curved as defined inRiemmanian geometry. Further, the minimum energy function has aformalism such that the Euler Lagrange operator of the minimum energyfunction yields tensor densities. As a result, the matching process isstable in curved spaces. This minimum energy function can further beused as computational basis for the development of new theories inphysics.

The modification to the minimum energy function used according to anembodiment of a method of the invention may be described as follows. InRiemannian geometry language the minimum energy to be minimized can bewritten as

$\begin{pmatrix}{{K_{1}P_{i}P^{i}} +} \\{{K_{2}( {\frac{( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} )( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} + \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}} )} +} \\{{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2} +} \\{K_{4}R}\end{pmatrix}\sqrt{\pm g}$Where: g is the determinant of the metric tensor, upper indices denotecontravariant tensor property, lower indices denote covariant tensorproperties, P_(i) denotes a gradient

$P_{i} \equiv \frac{\partial P}{\partial x^{i}}$of a gradually changing function P, That changes between the memory andinput layers, (−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d),(−d<x¹<d,−d<x²<d, . . . x^(j)+δ, . . . ,−d<x^(n)<d) (when n is thedimension of a Riemannian manifold and d>0 is a very large in relationto δ>0) Semi colon denotes a covariant derivative and comma denotes anordinary derivative,

$P_{i},_{j}{\equiv {\frac{\partial P_{i}}{\partial x^{j}}.}}$R denotes the scalar curvature also known as Ricci scalar. K₁, K₂, K₃,K₄ denote constants of the model.

The reason to include the curvature R as cost function is in order topenalize curvature which means that matching between patterns will notbe at any cost but rather will be also regulated when curvature alsocosts energy.

According to the earlier described embodiment, the energy function thatwas minimized was:

$\sum\limits_{k = 1}^{n}( {( {\sum\limits_{\lambda = 1}^{n}{( {\frac{\partial}{\partial x^{k}}P_{\lambda}} )P_{\lambda}}} )( {\sum\limits_{\lambda = 1}^{n}{( {\frac{\partial}{\partial x^{k}}P_{\lambda}} )P_{\lambda}}} )} )$and in full tensor language P^(s)P_(r);_(s)P^(s)P^(r);_(s). This minimumenergy function is applicable only in flat space and it's Euler Lagrangeoperator does not yield tensor densities.

A thorough explanation for the theory is as follows. As described abovein connection with the previous embodiment, the Far Span k−DistanceConnections has both digital and continuous forms. The continuous formof the energy described above may be expressed as

${\sum\limits_{i = 1}^{n}( \frac{\partial P}{\partial x^{i}} )^{2}} + {K{\sum\limits_{i = 1}^{n}( {\sum\limits_{j = 1}^{n}{\frac{\partial^{2}P}{{\partial x^{i}}{\partial x^{j}}}\frac{\partial P}{\partial x^{j}}}} )^{2}}}$when upper indices are x coordinate indices and not powers and n is thedimension of the compared memory and input patterns and K is constant orchanging in a relaxation process. In tensor form the function may bewritten as (P_(v)P^(v)+KP^(j)P^(i);_(j)P^(L)P_(i);_(L))√{square rootover (g)} when P^(j)P^(i);_(j)P^(L)P_(i);_(L) was the continuous localform of the Far Span k−Distance Connection. Semi colon denotes thecovariant derivative and indices of P denotes the pattern function thatis known on the memory layer and on the input layer. Also,

$P_{j} \equiv {\frac{\partial P}{\partial x^{j}}.}$P^(j)P^(i);_(j)P^(L)P_(i);_(L) has several problems significantlyincluding a numerical instability of the matching between the memory andinput layers.

To address these problems, we can define the Far Span k−DistanceConnection as a minimum energy function that depends on the second orderderivatives of a pattern P that is defined on two boundaries, with upperindices describing coordinate indices and not powers, boundary(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d) and boundary(−d<x¹<d,−d<x²<d, . . . x^(j)+δ, . . . ,−d<x^(n)<d) (when n is thedimension of a Riemannian manifold and d is very large in relation toδ).

This is in contrast to the prior definition of the domain,(−d<x¹<d,−d<x²<d, . . . ,−d<x^(n)<d) and the boundaries(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d), (−d<x¹<d,−d<x²<d, . .. x^(j)+δ, . . . ,−d<x^(n)<d) Referred to an Euclidean space, e.g. R³.

If one boundary (−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d)contains a pattern drawn as p and the other boundary has anotherdefinition of p we called (−d<x¹<d,−d<x²<d, . . . x^(j), . . .−d<x^(n)<d), memory layer and input layer. We then tried to connect thepattern p on both layers by the vector field which is generated by thegradient of this field p such that the gradient of p will form a vectorfield on the domain (−d<x¹<d,−d<x²<d, . . . ,−d<x^(n)<d) between thememory layer and the input layer and such that the gradient of p willform curves as parallel and as straight as possible.

Transformations between p that is defined on the input layer and p thatis defined on the memory layer need not be linear, e.g. a transformationbetween two hand written signatures defined on two parallel planes x,y,0and x,y,1 as p≠0 for ink dot and p=0 for blank paper.

The restriction of matching between such two patterns in Euclidean spacedoesn't make any sense because in curved space matching can be easier toperform.

The reason is that if parallel

${{Gradient}(p)} = \frac{\partial p}{\partial x^{i}}$for all i will not always be able to form parallel curves in Euclideanspaces but will be able to form parallel geodesic curves in curved spaceand thus allow a method by which a wider range transformations will becovered by the disclosed method.

The geometric motivation for extending the method may be explained asfollows. A problem is that, given two Gaussian coordinate boundaries,(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d), (−d<x¹<d,−d<x²<d, . .. x^(j)+δ, . . . ,−d<x^(n)<d) (when n is the dimension of a Riemannianmanifold and d is a very large in relation to δ) and a scalar field Pwhich its P² values are known on these boundaries, and so is thegeometry of the boundaries as geometric objects is also known, and0≦P²≦1, on each one of the boundaries, define a Lagrangian such that theintegral of (P^(i)P_(i))^(m) s.t.

$P_{i} = \frac{\partial P}{\partial x^{i}}$when the power m≧1 will be globally minimal under the conditions: P²≧ε²for some minimal value ε² and |P_(i)|²≦L², for some maximal value L².Here upper indices denote the contravariant property, lower ones thecovariant one and x^(i) denotes the coordinates. The problem is alsoknown as optimization of Homotopy. See, e.g., Victor Guillemin, AlanPollack Differential Topology, Homotopy and Stability, pages 33,34,35,ISBN 0-13-212605-2. Surprisingly, minimizing Lagrangians that involvedonly first order derivatives of the scalar field P did not accomplishthe required global minimum.

An interesting problem is to assign the root of the Ricci scalar (seeDavid Lovelock and Hanno Rund, Tensors, Differential Forms andVariational Principles 261, 3.26, ISBN 0-486-65840-6) to P or in patternrecognition language, to use the Ricci scalar to encode two comparedboundaries on which a geometric pattern is known. Intuitively speaking,the problem quite resembles comparing two hand written signatures suchthat each signature is etched (as the Support P≠0) on a two dimensionalplane and two planes with such signatures are parallel at distance δ onefrom the other, (−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d),(−d<x¹<d,−d<x²<d, . . . x^(j)+δ, . . . ,−d<x^(n)<d).

A good matching between these two signatures will be achieved by avector field

$P_{i} = \frac{\partial P}{\partial x^{i}}$that can be interpreted into curves connecting the two planes. If forexample one signature is an ellipse and the other one is a circle thenchecking the intersection of the field P_(i) with the intermediateplane,

$( {{{- d} < x^{1} < d},{{- d} < x^{2} < d},{{\ldots\mspace{14mu} x^{j}} + \frac{\delta}{2}},\ldots\mspace{11mu},{{- d} < x^{n} < d}} )$will be also an ellipse with eccentricity half the one of an ellipsewhich is defined by P≠0 on, (−d<x¹<d,−d<x²<d, . . . x^(j)+δ, . . .,−d<x^(n)<d).

Surprisingly, according to computer simulations, in order to achievesuch a field as a result of Euler Lagrange equations and of the calculusof variations, derivatives of the vector field P_(i) must be used. Usingthe well known covariant and contravariant indices notation of modernRiemannian Geometry, interesting candidate Lagrangians for explorationare

$L = {( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}\sqrt{g}}$when semi colon denotes the Covariant derivative and g denotes thedeterminant of the Metric Tensor. An alternative is

$L = {( {\frac{( {{P_{r}\text{;}_{s}} - {P_{s}\text{;}_{r}}} )( {{P_{L}\text{;}_{k}} - {P_{k}\text{;}_{L}}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} + \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}} ){\sqrt{g}.}}$The latter is suitable for yielding a meaningful theory also when P_(i)is not a gradient of some scalar field p when possibly

$\frac{( {{P_{r}\text{;}_{s}} - {P_{s}\text{;}_{r}}} )( {{P_{L}\text{;}_{k}} - {P_{k}\text{;}_{L}}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} \neq 0.$

As we found out in low dimensions, a stabilizing Lagrangian is the KGoperator P_(i)P^(i) and a meaningful theory is to minimize an integralof the form,

$\int_{\Omega}{\begin{pmatrix}{{K_{1}P_{i}P^{i}} +} \\{{K_{2}( {\frac{( {{P_{r}\text{;}_{s}} - {P_{s}\text{;}_{r}}} )( {{P_{L}\text{;}_{k}} - {P_{k}\text{;}_{L}}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} + \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}} )} +} \\{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2}\end{pmatrix}\sqrt{g}\ {\mathbb{d}\Omega}}$(see, U.S. patent application Ser. No. 10/144,754, Apparatus For AndMethod Of Pattern Recognition And Image Analysis) for some constants K₁,K₂, K₃.

When g denotes the determinant of the metric tensor g_(ij) and semicolon denotes the covariant derivative and comma denotes an ordinaryderivative as commonly used in differential geometry.

It appears that minimizing the term that involves only the field p_(i)is responsible for minimizing the field intensity between the twoboundaries and minimizing terms that involve derivatives of the field,such as P_(i);_(j) is responsible that the field p will not losegeometric structure as it changes from the boundary, (−d<x¹<d,−d<x²<d, .. . x^(j), . . . −d<x^(n)<d) to the boundary, (−d<x¹<d,−d<x²<d, . . .x^(j)+δ, . . . ,−d<x^(n)<d).

It seems that for fields that are the gradient of some scalar function(conserving fields) minimizing,

$\int_{\Omega}{( {{K_{1}P_{i}P^{i}} + {K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2}} )\sqrt{g}\ {\mathbb{d}\Omega}}$is sufficient.

The theory and method that will unfold is of purely geometric meaningand with typical Riemannian formalism. The Lagrangians result asdescribed below.

The Transformatron theory can further be explained as follows. Let usconsider a specific Lagrangian of the form

$\begin{matrix}{L( {g_{\mu\; v},\frac{\partial g_{\mu\; v}}{\partial x^{m}},P^{i},\frac{\partial P^{i}}{\partial x^{m}}} )} & (1)\end{matrix}$Such that x^(m) is our local coordinates system.g_(μv) is the metric tensor and P^(i) is a contravariant vector field.

From now on semi colon ‘;’ will 1 denote covariant derivative and comma‘,’ will denote ordinary derivative as in the following examples,

$P^{i},_{j}{\equiv \frac{\partial P^{i}}{\partial x^{j}}},$

${P^{i}\text{;}_{j}} \equiv {\frac{\partial P^{i}}{\partial x^{j}} + {{\Gamma_{k}^{i}}_{j}P^{k}}}$when Γ_(k) ^(i) _(j) are the Christoffel symbols

${\Gamma_{k}^{i}}_{j} \equiv {\frac{1}{2}{{g^{is}( {g_{{sk},j} + g_{{js},k} - g_{{kj},s}} )}.}}$

We can now write a simple Lagrangian

$\begin{matrix}{\begin{pmatrix}{{K_{2}( {\frac{( {{P_{r}\text{;}_{s}} - {P_{s}\text{;}_{r}}} )( {{P_{L}\text{;}_{k}} - {P_{k}\text{;}_{L}}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} + \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}} )} +} \\{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2}\end{pmatrix}\sqrt{g}} & (2)\end{matrix}$

The motivation for (2) is explained in further detail below.

Definition 1: The Lagrangian (2) is the Riemannian Far Span k DistanceConnection. The term

$L = {( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}\sqrt{g}}$turns out to be very useful in high dimensional pattern matching tasksin the field of analog computation and visual analysis.

Minimizing this term means that the square norm of the field P_(i) alongit's curves stays as stationary as possible. In directions perpendicularto the field P_(i), the field intensity is allowed to change. This iswhy (2.3) is a preferable Lagrangian for describing a field related tomatter. An example of a real problem is two parallel R² boards in R³ onwhich an image is encoded using an electric field.

One board encodes a memorized image and the other encodes a new input.The boards are divided into tiny pixels such that in the middle of eachpixel there is an electrode. The potential of the memorized image pixelsis positive and the potential of the pixels of the input image isnegative. Between the two boards needle shaped pieces of conductivematerial are suspended in liquid. These needle shaped pieces are coatedwith non-conductive spheres of polyethylene.

Since dipoles respond to the gradient of the square norm of an electricfield, the needles will align such that this gradient will be minimizedalong the direction they point to. L doesn't exactly describe thismachine but is basically designed as a result of the same generaltopological idea.

Please note that (2) uses first order derivatives of P_(i), providingP_(i)=√{square root over (ρ)}U_(i) for pattern density ρ.

$\begin{matrix}{{( {\,\frac{( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} )( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}}} )\sqrt{g}} = {( {\,\frac{( {P_{r},{\,_{s}{- P_{s}}},\,_{r}} )( {P_{L},{\,_{k}{- P_{k}}},\,_{L}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}}} )\sqrt{g}}} & (2.1)\end{matrix}$

Definition 2: The Lagrangian (2.1) is named Angular or Curl deviation.

$\begin{matrix}{L = {{\frac{{( {P^{\lambda}P_{\lambda}} )\,},_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}\sqrt{- g}} = {\frac{( {P_{\mu};{{{}_{}^{}{}_{}^{}} + {P_{\mu}P_{\lambda}}};\,_{m}} ){g^{\mu\lambda}( {P_{r};{{{}_{}^{}{}_{}^{}} + {P_{r}P_{s}}};\,_{k}} )}g^{rs}g^{mk}}{P^{i}P_{i}}\sqrt{g}}}} & (2.2)\end{matrix}$

Definition 3: The Lagrangain (2.2) is named, Forward Deviation.

Another form is Forward Deviation for Conserving Fields.

$\begin{matrix}{( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}\sqrt{g}} & (2.3)\end{matrix}$The equation

$\begin{matrix}{{\delta{\int\limits_{\Omega}{\begin{pmatrix}{{K_{1}P_{i}P^{i}} +} \\{{K_{2}( {\frac{\begin{matrix}( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} ) \\{( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}\end{matrix}}{P^{i}P_{i}} + \frac{\begin{matrix}{( {P^{\lambda}P_{\lambda}} ),_{m}} \\{( {P^{s}P_{s}} ),_{k}g^{mk}}\end{matrix}}{P^{i}P_{i}}} )} +} \\{{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2} +} \\{K_{4}R}\end{pmatrix}\sqrt{g}{\mathbb{d}\Omega}}}} = 0} & (2.4)\end{matrix}$is also referred to as a Transformatron.

The present embodiment can be summarized by the flow diagram of FIG. 17in which new step 1701 is performed following loading of the memory andinput patterns and is followed by use of vibrational waves as describedin the prior embodiment. A possible constraint is presented later. IfP_(i) is a gradient of some scalar function then in (2.4) K₂ can be 0.Conversely, if P_(i) is not a gradient of a scalar function then K₃ canbe 0.

In any case K₂=0 yields a more stable theory.

A difficult question is whether K₄=0 yields a flat geometry theory. K₂(or K₃) is a result of optimization of

${\int_{\Omega}{( {P^{i}P_{i}} )^{m}\sqrt{g}\ {\mathbb{d}\Omega}}},$for some m>0, by (2.3) and is determined by knowing K₁. g is thedeterminant of the metric tensor in space-time, R is the Ricci scalarfield and Ω is a domain of the form, (−d<x¹<d,−d<x²<d, . . . x^(j), . .. −d<x_(n)<d), (−d<x¹<d,−d<x²<d, . . . x^(j)+δ, . . . ,−d<x^(n)<d).

Clearly (2),(2.1),(2.2) are scalar densities (see, David Lovelock andHanno Rund, Tensors, Differential Forms and Variational Principles page113, 2.18, and the transformation law in page 114, 2.30, 4.2 TheNumerical Relative Tensors, ISBN 0-486-65840-6) Additional constraintsmay be also required to define a unique solution.

A thorough differential geometry discussion on how to reach (2.1) and(2.2) is included hereinbelow. However, a difficult and important openproblem is whether there is an operator of the form,

$\begin{matrix}{{L = {( {{\frac{1}{4}{K( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}} )}} + {P^{s}P_{s}}} )\sqrt{g}}}\mspace{11mu}{{{or}\mspace{14mu}{preferably}\mspace{14mu} L} = {( {{\frac{1}{4}{K( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2}} + {P^{s}P_{s}}} )\sqrt{g}}}} & (2.5)\end{matrix}$s.t. P=√{square root over (R)}, the root of Ricci scalar andP_(s)=(√{square root over (R)}),_(s), such that the forth order EulerLagrange operator of L will yield a tensor with a new physical meaning.Until now, attempts did not succeed to prove that there is such anon-trivial tensor in which third and fourth order derivatives of themetric tensor do not vanish. The significance of that open problem isdue to section 7 and the introduction because structure conservationcost and curvature may yield a purely metric classical theory. Thefailure is partly due to lack of clear mathematical theories onLagrangians with order of derivatives of the metric tensor which ishigher than 2. (2.4) is defined as a Transformatron problem if the Ricciscalar P=√{square root over (R)} is known on two three dimensionalboundaries, (−d<x¹<d,−d<x²<d,−d<x³<d,x⁴),(−d<x¹<d,−d<x²<d,−d<x³<d,x⁴+δ).

We proceed with the purpose of showing that the Euler Lagrange operatorof (2.4) yields tensor densities.

The forward deviation may be addressed as follows. We will calculateEuler Lagrange operators (see, David Lovelock and Hanno Rund, Tensors,Differential Forms and Variational Principles page 323, 5.2, CombinedVector-Metric Field Theory, Page 325, Remark 1, ISBN 0-486-65840-6) inorder to prove that they yield tensor density and thus the model is afeasible physical model.

$\begin{matrix}{{{{EU}\begin{pmatrix}L & g_{\mu\; v} & {g_{u\; v},_{m}}\end{pmatrix}} = {\frac{\partial L}{\partial g_{\mu\; v}} - {\frac{\mathbb{d}\;}{\mathbb{d}x^{m}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}}}}{L = {\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{z}P_{z}} ),_{k}}{P^{i}P_{i}}\sqrt{g}}}} & (3) \\{{\frac{\partial L}{{\partial g_{\mu\; v}},_{m}} = {{2( \frac{P^{\mu}P^{v}Z^{m}}{P^{i}P_{i}} )\sqrt{g}\mspace{14mu}{s.t.\mspace{14mu} Z^{m}}} = ( {P^{i}P_{i}} )}},_{j}g^{jm}} & (4)\end{matrix}$

We now calculate,

$\begin{matrix}{{{- \frac{\mathbb{d}}{\mathbb{d}x^{m}}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}} = {{{- 2}\frac{( {( {P^{\mu}P^{v}Z^{m}} );{{\,_{m}{- P^{i}}}P^{v}Z^{m}\Gamma_{i}^{\mu}{\,_{m}{- P^{\mu}}}P^{i}Z^{m}\Gamma_{i}^{v}\,_{m}}} )}{P^{i}P_{i}}\sqrt{g}} + {{+ 2}\frac{{( {P^{\mu}P^{v}Z^{m}} )( {P^{r}P_{r}} )},_{m}\sqrt{g}}{( {P^{i}P_{i}} )^{2}}}}} & (5)\end{matrix}$

In which the term−P^(i)P^(v)Z^(m)Γ_(i) ^(μ) _(m)−P^(μ)P^(i)Z^(m)Γ_(i) ^(v) _(m)  (6)spoils the tensor density character of (5).

Please note the following:

$\begin{matrix}{\frac{\partial\sqrt{g}}{\partial x^{m}} = {\Gamma_{m}^{i}{\,_{i}\sqrt{g}}}} & (7)\end{matrix}$

Please also note

$\begin{matrix}{{P^{\mu}P^{v}Z^{i}\Gamma_{i}^{m}{\,_{m}\sqrt{g}}} = {{P^{\mu}P^{v}Z^{m}\Gamma_{m}^{i}{\,_{i}\sqrt{g}}} = {P^{\mu}P^{v}Z^{m}\frac{\partial\sqrt{g}}{\partial x^{m}}}}} & (8)\end{matrix}$

Which led to the first term in (5). We continue calculating,

$\begin{matrix}{\frac{\partial L}{\partial g_{\mu\; v}} = {\begin{pmatrix}{{2\frac{{P^{\mu}\text{;}_{m}P^{v}Z^{m}} + {P^{\mu}P^{v}\text{;}_{m}Z^{m}} - {{\Gamma_{i}^{\mu}}_{m}P^{i}P^{v}Z^{m}} - {{\Gamma_{i}^{v}}_{m}P^{\mu}P^{i}Z^{m}}}{P^{i}P_{i}}} +} \\{\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}( {{g^{mv}g^{\mu\; k}} - {\frac{1}{2}g^{m\;\mu}g^{vk}}} )}{P^{i}P_{i}} +} \\{{{- \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{( {P^{i}P_{i}} )^{2}}}P^{\mu}P^{v}} +} \\{\frac{1}{2}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}g^{\mu\; v}}\end{pmatrix}\sqrt{g}}} & (9)\end{matrix}$

Please note

$\begin{matrix}{\frac{\partial g^{mk}}{\partial g^{\mu\; v}} = {{g^{mv}g^{\mu\; k}} - {\frac{1}{2}g^{m\;\mu}g^{vk}}}} & (10)\end{matrix}$

So the non-tensor components of (5) are the same as in (9) which assuresthat the following is a tensor density,

$\begin{matrix}{{\frac{\partial L}{\partial g_{\mu\; v}} - {\frac{\mathbb{d}}{\mathbb{d}x^{m}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}}} = {\begin{pmatrix}{{2\frac{{P^{\mu}\text{;}_{m}P^{v}Z^{m}} + {P^{\mu}P^{v}\text{;}_{m}Z^{m}} - {{\Gamma_{i}^{\mu}}_{m}P^{i}P^{v}Z^{m}} - {{\Gamma_{i}^{v}}_{m}P^{\mu}P^{i}Z^{m}}}{P^{i}P_{i}}} +} \\{\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}( {{g^{mv}g^{\mu\; k}} - {\frac{1}{2}g^{m\;\mu}g^{vk}}} )}{P^{i}P_{i}} +} \\{{{- \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{( {P^{i}P_{i}} )^{2}}}P^{\mu}P^{v}} +} \\{\frac{1}{2}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}g^{\mu\; v}} \\{{{- 2}\frac{( {{( {P^{\mu}P^{v}Z^{m}} )\text{;}_{m}} - {P^{i}P^{v}Z^{m}{\Gamma_{i}^{\mu}}_{m}} - {P^{\mu}P^{i}Z^{m}{\Gamma_{i}^{v}}_{m}}} )}{P^{i}P_{i}}} +} \\{{+ 2}\frac{{( {P^{\mu}P^{v}Z^{m}} )( {P^{r}P_{r}} )},_{m}}{( {P^{i}P_{i}} )^{2}}}\end{pmatrix}\sqrt{g}}} & (11)\end{matrix}$

In which the terms −P^(i)P^(v)Z^(m)Γ_(i) ^(μ) _(m)−P^(μ)P^(i)Z^(m)Γ_(i)^(v) _(m) cancel each other which clearly shows (11) is a tensordensity.

We continue calculating the other Euler Lagrange terms,

$\begin{matrix}{\frac{\partial L}{{\partial P_{\mu}},_{v}} = {2\frac{( {{P^{\mu}Z^{v}} + {P^{v}Z^{\mu}}} )}{P^{i}P_{i}}\sqrt{g}}} & (12) \\{{{{- \frac{\mathbb{d}}{\mathbb{d}x^{v}}}\frac{\partial L}{{\partial P_{\mu}},_{v}}} = {{{- 2}\frac{( {{( {{P^{\mu}Z^{v}} + {P^{v}Z^{\mu}}} )\text{;}_{v}} - {{\Gamma_{i}^{\mu}}_{v}P^{i}Z^{v}} - {{\Gamma_{i}^{\mu}}_{v}P^{v}Z^{i}}} )}{P^{i}P_{i}}\sqrt{g}} + {2\frac{( {{P^{\mu}Z^{v}} + {P^{v}Z^{\mu}}} )}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )}}},_{v}\sqrt{g}} & (13) \\{\frac{\partial L}{\partial P_{\mu}} = {{4\frac{( {{P^{\mu}\text{;}_{v}Z^{v}} - {{\Gamma_{i}^{\mu}}_{k}P^{i}Z^{k}}} )}{P^{i}P_{i}}\sqrt{g}} + {{- 2}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{( {P^{i}P_{i}} )^{2}}P^{\mu}\sqrt{g}}}} & (14)\end{matrix}$

Adding (14) and (13) we have

$\begin{matrix}{{{\frac{\partial L}{\partial P_{\mu}} - {\frac{\mathbb{d}}{\mathbb{d}x^{v}}\frac{\partial L}{{\partial P_{\mu}},_{v}}}} = {{4\frac{( {{P^{\mu}\text{;}_{v}Z^{v}} - {{\Gamma_{i}^{\mu}}_{k}P^{i}Z^{k}}} )}{P^{i}P_{i}}\sqrt{g}} + {{- 2}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{( {P^{i}P_{i}} )^{2}}P^{\mu}\sqrt{g}} - {2\frac{( {{( {{P^{\mu}Z^{v}} + {P^{v}Z^{\mu}}} )\text{;}_{v}} - {{\Gamma_{i}^{\mu}}_{v}P^{i}Z^{v}} - {{\Gamma_{i}^{\mu}}_{v}P^{v}Z^{i}}} )}{P^{i}P_{i}}\sqrt{g}} + {2\frac{( {{P^{\mu}Z^{v}} + {P^{v}Z^{\mu}}} )}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )}}},_{v}\sqrt{g}} & (15)\end{matrix}$

Clearly (15) is a tensor density because:−4Γ_(i) ^(μ) _(k) P ^(i) Z ^(k)=−2(Γ_(i) ^(μ) _(v) P ^(i) Z ^(v)+Γ_(i)^(μ) _(v) P ^(v) Z ^(i))  (16)

By (11) and (15) that we have proved, the Euler Lagrange operators of(2.2) is indeed a tensor density.

Angular/Curl deviation is addressed as follows. We will calculate theEuler Lagrange operators for

$L = {( \frac{( {P_{r},_{s}{- P_{s}},_{r}} )( {P_{L},_{k}{- P_{k}},_{L}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} )\sqrt{g}}$

Obviously

$\begin{matrix}{\frac{\partial L}{{\partial g_{\mu\; v}},_{m}} = 0} & (17)\end{matrix}$

We continue with

$\begin{matrix}{\frac{\partial L}{\partial g_{\mu\; v}} = {\frac{\partial}{\partial g_{\mu\; v}}( \frac{( {P_{r},_{s}{- P_{s}},_{r}} )( {P_{L},_{k}{- P_{k}},_{L}} )g^{rL}g^{sZ}g^{kU}P_{Z}P_{U}}{P^{i}P_{i}} )\sqrt{g}}} & (18)\end{matrix}$

We continue by calculating,

$\begin{matrix}{\frac{{\partial( {g^{rL}g^{sZ}g^{kU}} )}P_{Z}{P_{U}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )}{\partial g_{\mu\; v}} = {{{( {{g^{rv}g^{\mu\; L}} - {\frac{1}{2}g^{r\;\mu}g^{vL}}} )g^{sZ}g^{kU}P_{Z}{P_{U}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )} + {{g^{rL}( {{g^{sv}g^{\mu\; Z}} - {\frac{1}{2}g^{s\;\mu}g^{vZ}}} )}g^{kU}P_{Z}{P_{U}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )} + {g^{rL}{g^{sZ}( {{g^{kv}g^{\mu\; U}} - {\frac{1}{2}g^{k\;\mu}g^{vU}}} )}P_{Z}{P_{U}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )}} = {{( {{g^{rv}g^{\mu\; L}} - {\frac{1}{2}g^{r\;\mu}g^{vL}}} )P^{s}{P^{k}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )} + {( {{g^{sv}P^{\mu}} - {\frac{1}{2}g^{s\;\mu}P^{v}}} )P^{k}{g^{rL}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )} + {( {{g^{kv}P^{\mu}} - {\frac{1}{2}g^{k\;\mu}P^{v}}} )P^{s}{g^{rL}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )}}}} & (19)\end{matrix}$

So we can write,

$\begin{matrix}{{\frac{\partial L}{\partial g_{\mu\; v}} - {\frac{\mathbb{d}}{\mathbb{d}x^{m}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}}} = {\frac{\partial L}{\partial g_{\mu\; v}} = {{\begin{pmatrix}{{( {{g^{rv}g^{\mu\; L}} - {\frac{1}{2}g^{r\;\mu}g^{vL}}} )P^{s}{P^{k}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )} +} \\{{( {{g^{sv}P^{\mu}} - {\frac{1}{2}g^{s\;\mu}P^{v}}} )P^{k}{g^{rL}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )} +} \\{( {{g^{kv}P^{\mu}} - {\frac{1}{2}g^{k\;\mu}P^{v}}} )P^{s}{g^{rL}( {P_{r},_{s}{- P_{s}},_{r}} )}( {P_{L},_{k}{- P_{k}},_{L}} )}\end{pmatrix}\frac{\sqrt{g}}{P_{i}P^{i}}} + {{- ( \frac{( {P_{r},_{s}{- P_{s}},_{r}} )( {P_{L},_{k}{- P_{k}},_{L}} )g^{rL}g^{sZ}g^{kU}P_{Z}P_{U}}{( {P^{i}P_{i}} )^{2}} )}P^{\mu}P^{v}\sqrt{g}} + {\frac{1}{2}( \frac{( {P_{r},_{s}{- P_{s}},_{r}} )( {P_{L},_{k}{- P_{k}},_{L}} )g^{rL}g^{sZ}g^{kU}P_{Z}P_{U}}{P^{i}P_{i}} )g^{\mu\; v}\sqrt{g}}}}} & (20)\end{matrix}$

It will be recognized that (20) is a tensor density.

We continue by calculating

$\begin{matrix}{\frac{\partial L}{{\partial P_{u}},_{v}} = {{2( \frac{{F_{k}^{\mu}P^{v}P^{k}} - {F_{k}^{v}P^{\mu}P^{k}}}{P^{i}P_{i}} )\sqrt{g}\mspace{14mu}{s.t.\mspace{14mu} F_{\mu\; v}}} = ( {P_{\mu},_{v}{- P_{v}},_{\mu}} )}} & (21)\end{matrix}$

We may write

$\begin{matrix}{{2( \frac{{F_{k}^{\mu}P^{v}P^{k}} - {F_{k}^{v}P^{\mu}P^{k}}}{P^{i}P_{i}} )\sqrt{g}} = {2\frac{B^{\mu\; v}}{P^{i}P_{i}}\sqrt{g}}} & (22)\end{matrix}$

It should be apparent thatB ^(μv) =−B ^(vμ)  (23)which is of crucial importance as will be apparent.

We continue calculating,

$\begin{matrix}{{{- \frac{\mathbb{d}}{\mathbb{d}x^{m}}}\frac{\partial L}{{\partial P_{\mu}},_{v}}} = {{{- 2}( {B^{\mu\; v};{{\,_{v}{- B^{kv}}}\Gamma_{k}^{\mu}\,_{v}}} )\sqrt{g}} + {2( {{\frac{{F_{k}^{\mu}P^{v}P^{k}} - {F_{k}^{v}P^{\mu}P^{k}}}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )},_{v}} )\sqrt{g}}}} & (24)\end{matrix}$

By (23) we haveB^(kv)Γ_(k) ^(μ) _(v)=0  (25)

So (24) reduces to,

$\begin{matrix}{{{- \frac{\mathbb{d}}{\mathbb{d}x^{m}}}\frac{\partial L}{{\partial P_{\mu}},_{v}}} = {{{- 2}\frac{B^{\mu\; v};\,_{v}}{P^{i}P_{i}}\sqrt{g}} + {2( {{\frac{{F_{k}^{\mu}P^{v}P^{k}} - {F_{k}^{v}P^{\mu}P^{k}}}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )},_{v}} )\sqrt{g}}}} & (26)\end{matrix}$which is obviously a tensor density.

We continue calculating,

$\begin{matrix}{\frac{\partial L}{\partial P_{\mu}} = {{\frac{\partial}{\partial P_{\mu}}( \frac{F_{rs}F_{k}^{r}P_{z}P_{u}g^{sz}g^{ku}}{P^{i}P_{i}} )\sqrt{g}} = {{( \frac{{F_{r}^{\mu}F_{k}^{r}P^{k}} + {F_{rs}F^{r\;\mu}P^{s}}}{P^{i}P_{i}} )\sqrt{g}} + {{- 2}( \frac{F_{rs}F_{k}^{r}P_{z}g^{sz}g^{ku}}{( {P^{i}P_{i}} )^{2}} )P^{\mu}\sqrt{g}}}}} & (27)\end{matrix}$

Adding (27) and (26) we have,

$\begin{matrix}{{\frac{\partial L}{\partial P_{\mu}} - {\frac{\mathbb{d}}{\mathbb{d}x^{v}}\frac{\partial L}{{\partial P_{\mu}},_{v}}}} = {{\frac{\partial}{\partial P_{\mu}}( \frac{F_{rs}F_{k}^{r}P_{z}P_{u}g^{sz}g^{ku}}{P^{i}P_{i}} )\sqrt{g}} = {{( \frac{{F_{r}^{\mu}F_{k}^{r}P^{k}} + {F_{rs}F^{r\;\mu}P^{s}}}{P^{i}P_{i}} )\sqrt{g}} + {{- 2}( \frac{F_{rs}F_{k}^{r}P_{z}P_{u}g^{sz}g^{ku}}{( {P^{i}P_{i}} )^{2}} )P^{\mu}\sqrt{g}} + {{- 2}\frac{B^{\mu\; v};\,_{v}}{P^{i}P_{i}}\sqrt{- g}} + {2( {{\frac{{F_{k}^{\mu}P^{v}P^{k}} - {F_{k}^{v}P^{\mu}P^{k}}}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )},_{v}} )\sqrt{g}}}}} & (28)\end{matrix}$(28) is a tensor density. So we have proved that the Euler Lagrangeoperators of the Angular (or Curl) deviations yield tensor densities asrequired by physics. We therefore can consider (2.3) as a possibletheory.

To address the forward deviation for conserving fields we calculateEuler Lagrange operators (see, Lovelock et al., supra) in order to provethat they yield tensor density and thus the model is a feasible physicalmodel.

$\begin{matrix}{{{{EU}\begin{pmatrix}L & g_{\mu\; v} & {g_{\mu\; v},_{m}}\end{pmatrix}} = {\frac{\partial L}{\partial g_{\mu\; v}} - {\frac{\mathbb{d}}{\mathbb{d}x^{m}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}}}}{L = {( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}\sqrt{g}}}} & (29) \\{\frac{\partial L}{{\partial g_{\mu\; v}},_{m}} = {2( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{P^{\mu}P^{v}P^{m}}{P^{i}P_{i}}\sqrt{g}}} & (30)\end{matrix}$

We now calculate,

$\begin{matrix}{{{{- \frac{\mathbb{d}}{\mathbb{d}x^{m}}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}} = {{{- 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{ {( {P^{\mu}P^{v}P^{m}} );{{\,_{m}{- P^{i}}}P^{v}P^{m}\Gamma_{i}^{\mu}{\,_{m}{- P^{\mu}}}P^{i}P^{m}\Gamma_{i}^{v}\,_{m}}} )}{P^{i}P_{i}}\sqrt{g}} + {{- 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )}}},_{m}{{\frac{( {P^{\mu}P^{v}P^{m}} )}{P^{i}P_{i}}\sqrt{g}} + {{+ 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{{( {P^{\mu}P^{v}P^{m}} )( {P^{r}P_{r}} )},_{m}}{( {P^{i}P_{i}} )^{2}}\sqrt{g}}}} & (31)\end{matrix}$

In which the term−P^(i)P^(v)P^(m)Γ_(i) ^(μ) _(m)−P^(μ)P^(i)Z^(m)Γ_(i) ^(v) _(m)  (32)spoils the tensor density character of (31).

Please note the following:

$\begin{matrix}{\frac{\partial\sqrt{g}}{\partial x^{m}} = {\Gamma_{m}^{i}{\,_{i}\sqrt{g}}}} & (33)\end{matrix}$

Please also note,

$\begin{matrix}{{P^{\mu}P^{v}P^{i}\Gamma_{i}^{m}{\,_{m}\sqrt{g}}} = {{P^{\mu}P^{v}P^{m}\Gamma_{m}^{i}{\,_{i}\sqrt{g}}} = {P^{\mu}P^{v}P^{m}\frac{\partial\sqrt{g}}{\partial x^{m}}}}} & (34)\end{matrix}$

Which led to the first term in (31). We continue calculating,

$\begin{matrix}{\frac{\partial L}{\partial g_{\mu\; v}} = {\begin{pmatrix}{{2( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{\begin{matrix}{P^{\mu};{{{{}_{}^{}{}_{}^{}}P^{m}} + {P^{\mu}P^{v}}};{{{}_{}^{}{}_{}^{}} -}} \\{{\Gamma_{i}^{\mu}{{}_{}^{}{}_{}^{}}P^{v}P^{m}} - {\Gamma_{i}^{v}{{}_{}^{}{}_{}^{}}P^{i}P^{m}}}\end{matrix}}{P^{i}P_{i}}} +} \\{{{- 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{( {P^{i}P_{i}} )^{2}}P^{\mu}P^{v}} +} \\{\frac{1}{2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}g^{\mu\; v}}\end{pmatrix}\sqrt{g}}} & (35)\end{matrix}$

So the non-tensor components of (31) are the same as in (35) whichassures that the following is a tensor density,

$\begin{matrix}{{\frac{\partial L}{\partial g_{\mu\; v}} - {\frac{\mathbb{d}}{\mathbb{d}x^{m}}\frac{\partial L}{{\partial g_{\mu\; v}},_{m}}}} = {\begin{pmatrix}{{2( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{\begin{matrix}{P^{\mu};{{{{}_{}^{}{}_{}^{}}P^{m}} + {P^{\mu}P^{v}}};{{{}_{}^{}{}_{}^{}} -}} \\{{\Gamma_{i}^{\mu}{{}_{}^{}{}_{}^{}}P^{v}P^{m}} - {\Gamma_{i}^{v}{{}_{}^{}{}_{}^{}}P^{i}P^{m}}}\end{matrix}}{P^{i}P_{i}}} +} \\{{{- 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{( {P^{i}P_{i}} )^{2}}P^{\mu}P^{v}} +} \\{{\frac{1}{2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}g^{\mu\; v}} +} \\{{{- 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{\begin{matrix}{( {P^{\mu}P^{v}P^{m}} );{{\,_{m}{- P^{i}}}P^{v}P^{m}\Gamma_{i}^{\mu}{\,_{m} -}}} \\ {P^{\mu}P^{i}P^{m}\Gamma_{i}^{v}\,_{m}} )\end{matrix}}{P^{i}P_{i}}} +} \\{{{- 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )},_{m}{\frac{( {P^{\mu}P^{v}P^{m}} )}{P^{i}P_{i}} +}} \\{{+ 2}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{{( {P^{\mu}P^{v}P^{m}} )( {P^{r}P_{r}} )},_{m}}{( {P^{i}P_{i}} )^{2}}}\end{pmatrix}\sqrt{g}}} & (36)\end{matrix}$

In which the terms −P^(i)P^(v)P^(m)Γ_(i) ^(μ) _(m)−P^(μ)P^(i)P^(m)Γ_(i)^(v) _(m) cancel each other which clearly shows (36) is a tensordensity.

We continue calculating the other Euler Lagrange terms,

$\begin{matrix}{\frac{\partial L}{{\partial P_{\mu}},_{v}} = {4( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{P^{\mu}P^{v}}{P^{i}P_{i}}\sqrt{g}}} & (37) \\{{{{- \frac{\mathbb{d}}{\mathbb{d}x^{v}}}\frac{\partial L}{{\partial P_{\mu}},_{v}}} = {{{- 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\mu}P^{v}} );{{\,_{v}{- \Gamma_{i}^{\mu}}}{{}_{}^{}{}_{}^{}}P^{v}}}{P^{i}P_{i}}\sqrt{g}} + {{- 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )}}},_{v}{{\frac{P^{\mu}P^{v}}{P^{i}P_{i}}\sqrt{g}} + {{+ 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\mu}P^{v}} )}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )}},_{v}\sqrt{g}} & (38) \\{\frac{\partial L}{\partial P_{\mu}} = {{4( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\mu};{{{}_{}^{}{}_{}^{}} - {\Gamma_{i}^{\mu}{{}_{}^{}{}_{}^{}}P^{k}}}} )}{P^{i}P_{i}}\sqrt{g}} + {{- 4}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{( {P^{i}P_{i}} )^{2}}P^{\mu}\sqrt{g}}}} & (39)\end{matrix}$

Adding (38) and (39) we have

$\begin{matrix}{{{\frac{\partial L}{\partial P_{\mu}} - {\frac{\mathbb{d}}{\mathbb{d}x^{v}}\frac{\partial L}{{\partial P_{\mu}},_{v}}}} = {{{+ 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\mu};{{{}_{}^{}{}_{}^{}} - {\Gamma_{i}^{\mu}{{}_{}^{}{}_{}^{}}P^{k}}}} )}{P^{i}P_{i}}\sqrt{g}} + {{- 4}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{( {P^{i}P_{i}} )^{2}}P^{\mu}\sqrt{g}} + {{- 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\mu}P^{v}} );{{\,_{v}{- \Gamma_{i}^{\mu}}}{{}_{}^{}{}_{}^{}}P^{v}}}{P^{i}P_{i}}\sqrt{g}} + {{- 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )}}},_{v}{{\frac{P^{\mu}P^{v}}{P^{i}P_{i}}\sqrt{g}} + {{+ 4}( \frac{( {P^{\lambda}P_{\lambda}} ),_{s}P^{s}}{P^{i}P_{i}} )\frac{( {P^{\mu}P^{v}} )}{( {P^{i}P_{i}} )^{2}}( {P^{i}P_{i}} )}},_{v}\sqrt{g}} & (40)\end{matrix}$

It is apparent that (40) is a tensor density because the non-tensorterms cancel each other.

The motivation for the Lagrangian presented in (2) above may beexplained by first providing a definition for the term “Transformatron”.In particular, the geometric meaning of the “Transformatron” may bedescribed as follows. According to one explanation, the Transformatronfunctions by “penalizing” the Lagarngian for a vector field which is notgeodesic, can yield a global minimum of another integral under certainrestrictions. This penalty (or cost function) in turn, can take part ofthe “job” that Ricci tensor does, which means that our vector field willnot have to be geodesic but deviation from a geodesic field will have tobe “penalized”. We bear in mind that the term “job” refers to a deepDifferential Topology issue which is known as Stable Homotopy. See,e.g., Guillemin et al., supra). Moreover, the original Transformatronwas an image processing device and was implemented as a complextriangular grid structure and space was not considered as a continuum.

We continue by repeating the following, suppose we have a scalar fieldP_(i) who's values are known on the Gaussian coordinates boundaries,(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x_(n)<d), (−d<x¹<d,−d<x²<d . . .. x^(j)+δ, . . . ,−d<x^(n)<d).

Now we want to minimize P_(i)P^(i) on one hand and keep the field asgeodesic as possible in flat geometry. We assume that the Euler Number(see, John W. Milnor, Topology from the Differentiable Viewpoint, pages32-41, ISBN 0-691-04833-9) of our vector field P_(i) is 0 in every pointof the vacuo. This is a necessary condition for the following techniqueto work ! If our field P_(i) is smooth and doesn't posses catastrophicEuler Numbers, then differential geometry tells us that there exists anarc length curve x^(i)(s) such that

$\begin{matrix}{\frac{\mathbb{d}{x^{i}(s)}}{\mathbb{d}s} = \frac{P^{i}}{\sqrt{P^{L}P_{L}}}} & (41)\end{matrix}$

Such a curve is geodesic only if

$\begin{matrix}{\frac{\mathbb{d}^{2}{x^{i}(s)}}{\mathbb{d}s^{2}} = 0} & (42) \\{{\frac{\mathbb{d}^{2}{x^{i}(s)}}{\mathbb{d}s^{2}} = ( \frac{P^{i}}{\sqrt{P^{L}P_{L}}} )};{{{\,_{v}\frac{\mathbb{d}}{\mathbb{d}s}}x^{v}} = ( \frac{P^{i}}{\sqrt{P^{L}P_{L}}} )};{{\,_{v}\frac{P^{v}}{\sqrt{P^{L}P_{L}}}} = {{( {{\frac{P^{i};\,_{v}}{( {P^{L}P_{L}} )^{\frac{1}{2}}} - {\frac{1}{2}\frac{P^{i}}{( {P^{L}P_{L}} )^{\frac{3}{2}}}( {P^{L}P_{L}} )}};\,_{v}} )\frac{P^{v}}{( {P^{L}P_{L}} )^{\frac{1}{2}}}} = {( {{\frac{P^{i};{{}_{}^{}{}_{}^{}}}{( {P^{L}P_{L}} )} - {\frac{1}{2}\frac{P^{i}}{( {P^{L}P_{L}} )^{2}}( {P^{L}P_{L}} )}};{{}_{}^{}{}_{}^{}}} ) = 0}}}} & (43)\end{matrix}$

If our geodesic field P_(i) fulfills the following,P_(i);_(j)=P_(j);_(i) or in other words, is a conserving field, then itis sufficient for the norm of the field to be stationary along the curvex^(i)(s) because then(P _(i) P _(j) g ^(ij));_(m)=(P _(i);_(m) P ^(i) +P _(j);_(m) P ^(j))=0

P _(i);_(m) P ^(i)=0  (44)by which it is obvious that (43) vanishes.

If we want to construct a cost function term that relies solely on thederivatives of the square norm of a vector field then we will have aproblem when the field is not conserving.

One elegant way to solve this problem is to look at the following terms,

$\begin{matrix}{V^{r} = {( {P_{r};{\,_{s}{+ P_{s}}};\,_{r}} )\frac{P^{s}}{\sqrt{P^{i}P_{i}}}}} & (45) \\{U^{r} = {{( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} )\frac{P^{s}}{\sqrt{P^{i}P_{i}}}} = {( {P_{r},_{s}{- P_{s}},_{r}} )\frac{P^{s}}{\sqrt{P^{i}P_{i}}}}}} & (46)\end{matrix}$(2.1) is simply U_(r)U^(r)√{square root over (−g)} and (2.2) is simplyV_(r)V^(r)√{square root over (−g)}.

In the past I used to think that

$\begin{matrix}{( \frac{( {P_{r};{\,_{s}{+ P_{s}}};\,_{r}} )P^{r}P^{s}}{P^{i}P_{i}} )^{2}\sqrt{- g}} & (47)\end{matrix}$or its identical form,

$\begin{matrix}{( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )^{2}\sqrt{- g}} & (48)\end{matrix}$are preferable tensor densities because (48) has a clear topologicalmeaning and is more stable in pattern recognition engineeringapplications.

If for example, P_(i)=√{square root over (ρ)}U_(i) for some scalar fieldρ then 48 is not sensitive to gradiends of ρ which are perpendicular tothe field P_(i) that is if

${\frac{P_{i}}{2\sqrt{\rho}}\rho^{i}} = 0.$

This is a very useful property because a gradient ρ^(i) can representstructural information and forcing it to 0 causes this information to belost.

The problem is that

$\begin{matrix}{{( \frac{( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} )P^{r}P^{s}}{P^{i}P_{i}} )^{2}\sqrt{- g}} \equiv 0} & (49)\end{matrix}$which is not useful if P_(i) is not a conserving field. (2.1) solvesthat problem.

The problem that was left was to show that the Euler Lagrange operatorsof (2.1) (2.2) and (2.3) are indeed tensor densities. That is a criticalcondition for considering such a theory to describe Nature.

We can thus conclude that Stable Homotopy need not require geodesics toexist but does require to minimize deviations from the geodesicsproperties of the field that describes matter in vacuo.

As described, the Energy function which is also described herinabove asFar Span k−Distance Connection can be extended to Riemannian geometry inthe example forms 2.4 and 2.5. Further, the system referred to as theTransformatron can be numerically simulated such that the memory layer,the input layer and the domain in between will all be described ascoordinates of Riemannian manifold and thus allow convergence fordifficult transformations between the pattern stored in the input layerand the pattern that is stored in the memory layer.

The above referenced conservation constraint may be explained asfollows. A natural constraint relates to the conservation of matter andcan be

$\begin{matrix}{{\int_{\Omega}{\frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}}\sqrt{- g}\ {\mathbb{d}\Omega}}} = 0} & (A)\end{matrix}$

Adding the constraint (A) to (2) we have

$\begin{matrix}{{\delta{\int_{\Omega}{\begin{pmatrix}{{K_{l}P_{i}P^{i}} +} \\{{K_{2}( {\frac{\begin{matrix}( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} ) \\{( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}\end{matrix}}{P^{i}P_{i}} + \frac{\begin{matrix}{( {P^{\lambda}P_{\lambda}} ),_{m}} \\{( {P^{s}P_{s}} ),_{k}g^{mk}}\end{matrix}}{P^{i}P_{i}}} )} +} \\{R +} \\{{{+ ( {K_{3} + \lambda} )}\frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}}} + {K_{4}R}}\end{pmatrix}\sqrt{- g}{\mathbb{d}\Omega}}}} = 0} & (B)\end{matrix}$

(A) along with (B) define a constrained variational equation.

In this case the constant λ can be chosen to be −K₃ and yield a theoryin which a non-conserving field is not lost or added to vacuo.

The present application includes a significant amount of theory andderivation of equations. This information is included to assist oneskilled in the art in understanding the invention in detail. Theinclusion of this theory and these derivations are not intended orincluded to limit the present invention.

It should be noted and understood that all publications, patents andpatent applications mentioned in this specification are indicative ofthe level of skill in the art to which the invention pertains. Allpublications, patents and patent applications are herein incorporated byreference to the same extent as if each individual publication, patentor patent application was specifically and individually indicated to beincorporated by reference in its entirety.

1. A method of comparing an input pattern with a memory patterncomprising the steps of: loading a representation of said input patterninto cells in an input layer; loading a representation of said memorypattern into cells in a memory layer; loading an initial value intocells in an intermediate layers between said input layer and said memorylayer; comparing values of cells in said intermediate layers with valuesstored in cells of adjacent layers including calculating a minimumenergy connection in curved space from said cell in said intermediatelayer to said cells in said intermediate layers; updating values storedin cells in said intermediate layers based on said step of comparing;and mapping cells in said memory layer to cells in said input layer. 2.The method of claim 1 wherein said curved space is defined in Riemmaniangeometry.
 3. The method of claim 1 wherein the minimum energy is writtenas: $\begin{pmatrix}{{K_{l}P_{i}P^{i}} +} \\{{K_{2}( {\frac{\begin{matrix}( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} ) \\{( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}\end{matrix}}{P^{i}P_{i}} + \frac{\begin{matrix}{( {P^{\lambda}P_{\lambda}} ),_{m}} \\{( {P^{s}P_{s}} ),_{k}g^{mk}}\end{matrix}}{P^{i}P_{i}}} )} +} \\{{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2} +} \\{K_{4}R}\end{pmatrix}\sqrt{\pm g}$ where: g is the determinant of the metrictensor, upper indices denote contravariant tensor property, lowerindices denote covariant tensor properties, P_(i) denotes a gradient$P_{i} \equiv \frac{\partial P}{\partial x^{i}}$ of a gradually changingfunction P, that changes between the memory and input layers,(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d), (−d<x¹<d,−d<x²<d, . .. x^(j)+δ, . . . ,−d<x^(n)<d) (when n is the dimension of a Riemannianmanifold and d>0 is a very large in relation to δ>0), semi colon denotesa covariant derivative and comma denotes an ordinary derivative,$P_{i},_{j}{\equiv {\frac{\partial P_{i}}{\partial x^{j}}.}}$ R denotesthe scalar curvature also known as Ricci scalar, and K₁, K₂, K₃, K₄denote constants of the model.
 4. The method of claim 1 furtherincluding a step of: ascertaining said representation of said inputpattern by preprocessing said input pattern.
 5. The method of claim 1further including a step of: ascertaining said representation of saidmemory pattern by preprocessing said memory pattern.
 6. The method ofclaim 1 wherein at least one of said initial values loaded into saidcells in said intermediate layers are identical to said representationof said input pattern.
 7. The method of claim 1 wherein at least one ofsaid initial values loaded into said cells in said intermediate layersare identical to said representation of said memory pattern.
 8. Themethod of claim 1 wherein said step of comparison includes comparingsaid value in said cell with values in three adjacent cells in saidadjacent layer.
 9. The method of claim 1 wherein said step of comparisonincludes calculating a distance in curved space from said cell in saidintermediate layer to said cells in said intermediate layers.
 10. Themethod of claim 1 wherein said step of mapping cells includes the stepsof: introducing a vibration wave into a cell of said input layer.
 11. Amethod of comparing an input pattern with a memory pattern comprisingthe steps of: loading a representation of said input pattern into cellsin an input layer; loading a representation of said memory pattern intocells in a memory layer; loading an initial value into cells in anintermediate layers between said input layer and said memory layer;comparing values of cells in said intermediate layers with values storedin cells of adjacent layers through curved space; updating values storedin cells in said intermediate layers based on said step of comparing;and mapping cells in said memory layer to cells in said input layerincluding introducing a vibration wave into a cell of said input layer.12. The method of claim 11 wherein said curved space is defined inRiemmanian geometry.
 13. The method of claim 11 wherein said step ofcomparing includes a step of calculating a minimum energy connection insaid curved space from said cells in said intermediate layer to saidcells in said intermediate layers.
 14. The method of claim 13 whereinthe minimum energy is written as: $\begin{pmatrix}{{K_{l}P_{i}P^{i}} +} \\{{K_{2}( {\frac{\begin{matrix}( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} ) \\{( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}\end{matrix}}{P^{i}P_{i}} + \frac{\begin{matrix}{( {P^{\lambda}P_{\lambda}} ),_{m}} \\{( {P^{s}P_{s}} ),_{k}g^{mk}}\end{matrix}}{P^{i}P_{i}}} )} +} \\{{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2} +} \\{K_{4}R}\end{pmatrix}\sqrt{\pm g}$ where: g is the determinant of the metrictensor, upper indices denote contravariant tensor property, lowerindices denote covariant tensor properties, P_(i) denotes a gradient$P_{i} \equiv \frac{\partial P}{\partial x^{i}}$ of a gradually changingfunction P, that changes between the memory and input layers,(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d), (−d<x¹<d,−d<x²<d, . .. x^(j)+δ, . . . ,−d<x^(n)<d) (when n is the dimension of a Riemannianmanifold and d>0 is a very large in relation to δ>0), semi colon denotesa covariant derivative and comma denotes an ordinary derivative,$P_{i},_{j}{\equiv {\frac{\partial P_{i}}{\partial x^{j}}.}}$ R denotesthe scalar curvature also known as Ricci scalar, and K₁, K₂, K₃, K₄denote constants of the model.
 15. The method of claim 11 furtherincluding a step of: ascertaining said representation of said inputpattern by preprocessing said input pattern.
 16. The method of claim 11further including a step of: ascertaining said representation of saidmemory pattern by preprocessing said memory pattern.
 17. The method ofclaim 11 wherein at least one of said initial values loaded into saidcells in said intermediate layers are identical to said representationof said input pattern.
 18. The method of claim 11 wherein at least oneof said initial values loaded into said cells in said intermediatelayers are identical to said representation of said memory pattern. 19.The method of claim 11 wherein said step of comparison includescomparing said value in said cell with values in three adjacent cells insaid adjacent layer.
 20. The method of claim 11 wherein said step ofcomparison includes calculating a distance in curved space from saidcell in said intermediate layer to said cells in said intermediatelayers.
 21. An apparatus for comparing an input pattern with a memorypattern, the apparatus comprising: means for loading a representation ofsaid input pattern into cells in an input layer; means for loading arepresentation of said memory pattern into cells in a memory layer;means for loading an initial value into cells in an intermediate layersbetween said input layer and said memory layer; means for comparingvalues of cells in said intermediate layers with values stored in cellsof adjacent layers including calculating a minimum energy connection incurved space from said cell in said intermediate layer to said cells insaid intermediate layers; means for updating values stored in cells insaid intermediate layers based on said step of comparing; and means formapping cells in said memory layer to cells in said input layer.
 22. Theapparatus of claim 21 wherein said curved space is defined in Riemmaniangeometry.
 23. The method of claim 21 wherein the minimum energy iswritten as: $\begin{pmatrix}{{K_{l}P_{i}P^{i}} +} \\{{K_{2}( {\frac{\begin{matrix}( {P_{r};{\,_{s}{- P_{s}}};\,_{r}} ) \\{( {P_{L};{\,_{k}{- P_{k}}};\,_{L}} )g^{rL}P^{s}P^{k}}\end{matrix}}{P^{i}P_{i}} + \frac{\begin{matrix}{( {P^{\lambda}P_{\lambda}} ),_{m}} \\{( {P^{s}P_{s}} ),_{k}g^{mk}}\end{matrix}}{P^{i}P_{i}}} )} +} \\{{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2} +} \\{K_{4}R}\end{pmatrix}\sqrt{\pm g}$ where: g is the determinant of the metrictensor, upper indices denote contravariant tensor property, lowerindices denote covariant tensor properties, P_(i) denotes a gradient$P_{i} \equiv \frac{\partial P}{\partial x^{i}}$ of a gradually changingfunction P, that changes between the memory and input layers,(−d<x¹<d,−d<x²<d, . . . x^(j), . . . −d<x^(n)<d), (−d<x¹<d,−d<x²<d, . .. x^(j)+δ, . . . ,−d<x^(n)<d) (when n is the dimension of a Riemannianmanifold and d>0 is a very large in relation to δ>0), semi colon denotesa covariant derivative and comma denotes an ordinary derivative,$P_{i,j} \equiv {\frac{\partial P_{i}}{\partial x^{j}}.}$ R denotes thescalar curvature also known as Ricci scalar, and K₁, K₂, K₃, K₄ denoteconstants of the model.
 24. The apparatus of claim 21 further includingmeans for ascertaining said representation of said input pattern bypreprocessing said input pattern.
 25. The apparatus of claim 21 furtherincluding means for ascertaining said representation of said memorypattern by preprocessing said memory pattern.
 26. The apparatus of claim21 wherein at least one of said initial values loaded into said cells insaid intermediate layers are identical to said representation of saidinput pattern.
 27. The apparatus of claim 21 wherein at least one ofsaid initial values loaded into said cells in said intermediate layersare identical to said representation of said memory pattern.
 28. Theapparatus of claim 21 wherein means for comparing includes means forcomparing said value in said cell with values in three adjacent cells insaid adjacent layer.
 29. The apparatus of claim 21 wherein said meansfor comparing includes means for calculating a distance in curved spacefrom said cell in said intermediate layer to said cells in saidintermediate layers.
 30. The apparatus of claim 21 wherein said meansfor mapping cells includes means for introducing a vibration wave into acell of said input layer.
 31. An apparatus for comparing an inputpattern with a memory pattern, the apparatus comprising: means forloading a representation of said input pattern into cells in an inputlayer; means for loading a representation of said memory pattern intocells in a memory layer; means for loading an initial value into cellsin an intermediate layers between said input layer and said memorylayer; means for comparing values of cells in said intermediate layerswith values stored in cells of adjacent layers comparing values of cellsin said intermediate layers with values stored in cells of adjacentlayers through curved space; means for updating values stored in cellsin said intermediate layers in response to said means for comparing; andmeans for mapping cells in said memory layer to cells in said inputlayer including introducing a vibration wave into a cell of said inputlayer.
 32. The apparatus of claim 31 wherein said curved space isdefined in Riemmanian geometry.
 33. The apparatus of claim 31 whereinsaid means for comparing includes means for calculating a minimum energyconnection in said curved space from said cells in said intermediatelayer to said cells in said intermediate layers.
 34. The apparatus ofclaim 33 wherein the minimum energy is written as: $\begin{pmatrix}{{K_{1}P_{i}P^{i}} +} \\{{K_{2}( {\frac{( {P_{r;s} - P_{s;r}} )( {P_{L;k} - P_{k;L}} )g^{rL}P^{s}P^{k}}{P^{i}P_{i}} + \frac{( {P^{\lambda}P_{\lambda}} ),_{m}( {P^{s}P_{s}} ),_{k}g^{mk}}{P^{i}P_{i}}} )} +} \\{{K_{3}( \frac{( {P^{\lambda}P_{\lambda}} ),_{m}P^{m}}{P^{i}P_{i}} )}^{2} + {K_{4}R}}\end{pmatrix}\sqrt{\pm g}$ where: g is the determinant of the metrictensor, upper indices denote contravariant tensor property, lowerindices denote covariant tensor properties, P_(i) denotes a gradient$P_{i} \equiv \frac{\partial P}{\partial x^{i}}$ of a gradually changingfunction P, that changes between the memory and input layers,(−d<x¹<d,−d<x²<d, . . . −d<x^(n)<d), (−d<x¹<d,−d<x²<d, . . . x^(j)+δ, .. . ,−d<x^(n)<d) (when n is the dimension of a Riemannian manifold andd>0 is a very large in relation to δ>0), semi colon denotes a covariantderivative and comma denotes an ordinary derivative,$P_{i,j} \equiv {\frac{\partial P_{i}}{\partial x^{j}}.}$ R denotes thescalar curvature also known as Ricci scalar, and K₁, K₂, K₃, K₄ denoteconstants of the model.
 35. The apparatus of claim 31 further includingmeans for ascertaining said representation of said input pattern bypreprocessing said input pattern.
 36. The apparatus of claim 31 furtherincluding means for ascertaining said representation of said memorypattern by preprocessing said memory pattern.
 37. The apparatus of claim31 wherein at least one of said initial values loaded into said cells insaid intermediate layers are identical to said representation of saidinput pattern.
 38. The apparatus of claim 31 wherein at least one ofsaid initial values loaded into said cells in said intermediate layersare identical to said representation of said memory pattern.
 39. Theapparatus of claim 31 wherein said means for comparing includes meansfor comparing said value in said cell with values in three adjacentcells in said adjacent layer.
 40. The apparatus of claim 31 wherein saidmeans for comparing includes means for calculating a Euclidean distancefrom said cell in said intermediate layer to said cells in saidintermediate layers.